Adding polynomials in vertical and horizontal formats, subtracting by adding the opposite, multiplying two and three polynomials, the sum-and-difference, square and cube patterns, and multiplying two models to build a third.
Subject: Algebra 2 · 65 slides · symbolic lesson
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Title
Algebra 2 · Chapter 5 — Polynomials and Polynomial Functions
Add, Subtract, and Multiply Polynomials
Objectives
Five outcomes. The last one is why the arithmetic is worth having.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 346-351 — the lesson these objectives are drawn from
Warm-up
Lesson 1.2 combined like terms and Lesson 4.4 multiplied two binomials. Nothing here is new in kind.
Discussion prompt
Multiply the quantity x plus 3 by the quantity 3x squared minus 2x plus 4. How many individual products will you need, and how do you know you have them all?
Hint: Count the terms in each bracket.
Answer:
\[ (x+3)(3x^2 - 2x + 4) = 3x^3 + 7x^2 - 2x + 12 \]
Two terms times three terms is six products. FOIL was the special case of two by two, giving four; the general rule is that each term of the first bracket multiplies each term of the second, and the count is the product of the two term counts.
Concept
Adding and subtracting polynomials means combining coefficients of terms with the same power, which is Lesson 1.2's like-terms rule. Multiplying means pairing every term of one polynomial with every term of the other, which is the distributive property applied repeatedly.
like terms — Terms with the same variables raised to the same exponents. Only like terms can be combined by adding or subtracting their coefficients.
\[ (a+b)(c+d+e) = ac+ad+ae+bc+bd+be \]
The two operations have opposite difficulties. Adding is easy but the terms must be sorted first; multiplying needs no sorting but every pair must be found.
Figure (svg): Two columns separating pairs of terms that combine from pairs that do not
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 346-347
Section
Section 1
Concept
To add polynomials, add the coefficients of terms with the same power. A vertical format aligns like terms in columns, leaving a gap for any missing power; a horizontal format gathers like terms in a line before combining them.
\[ (3y^3 - 2y^2 - 7y) + (-4y^2 + 2y - 5) = 3y^3 - 6y^2 - 5y - 5 \]
Neither format is better. The vertical one makes a missing power visible as an empty column, which is useful for long polynomials; the horizontal one is faster for short ones.
Figure (svg): The same polynomial addition performed in a vertical column format and in a horizontal line format
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 346-346 — Add polynomials vertically and horizontally
Picture it
Example 1: one addition in each format.
Figure (svg): The same polynomial addition performed in a vertical column format and in a horizontal line format
In the vertical example the second polynomial has no x term, so a gap holds its place. Filling that gap with a zero, as in Lesson 5.2's tableau, works just as well.
Worked example
Example 1, parts a and b.
\[ \text{Add } 2x^3 - 5x^2 + 3x - 9 \text{ and } x^3 + 6x^2 + 11 \text{ vertically.} \]
Write one above the other, aligned by power
Why: The second has no x term, so that column is left empty.
Add the cubic column
Why: Two x cubed plus x cubed is 3x cubed.
\[ 3 x ^{3} \]
Add the quadratic and linear columns
Why: Negative 5x squared plus 6x squared is x squared; the x column has only 3x.
\[ x ^{2} + 3 x \]
Add the constants
Why: Negative 9 plus 11 is 2.
\[ +2 \]
Figure (svg): The solution to Worked example add in both formats shown as a ladder of expressions, one row per algebraic move
\[ 3x^3 + x^2 + 3x + 2 \]
Verify: evaluate both sides at a convenient value
Why: At x equal to 1 the two polynomials are 2 minus 5 plus 3 minus 9, or negative 9, and 1 plus 6 plus 11, or 18; their sum is 9. The answer at x equal to 1 is 3 plus 1 plus 3 plus 2, which is 9. Substituting a single value catches most arithmetic slips in a sum.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 346-346
Sorting
Same variables, same exponents.
Sort into buckets
Sort each pair.
The second unlike pair is the instructive one: x squared y and x y squared use the same letters but are not interchangeable, as substituting x equal to 2 and y equal to 3 shows.
Worked example
Example 1b and Guided Practice 1.
\[ \text{Add } (3y^3 - 2y^2 - 7y) + (-4y^2 + 2y - 5) \text{ and } (t^2 - 6t + 2) + (5t^2 - t - 8). \]
First: gather like terms
Why: The cubic term is alone; the squared terms are negative 2 and negative 4; the linear terms are negative 7 and 2.
First: combine
Why: Negative 2 minus 4 is negative 6; negative 7 plus 2 is negative 5; the constant is negative 5.
\[ 3 y ^{3} - 6 y ^{2} - 5 y - 5 \]
Second: gather
Why: The squared terms are 1 and 5; the linear terms are negative 6 and negative 1; the constants are 2 and negative 8.
Second: combine
Why: One plus 5 is 6; negative 6 minus 1 is negative 7; 2 minus 8 is negative 6.
\[ 6 t ^{2} - 7 t - 6 \]
Figure (svg): The solution to Worked example two more sums shown as a ladder of expressions, one row per algebraic move
\[ 3y^3 - 6y^2 - 5y - 5, \qquad 6t^2 - 7t - 6 \]
Verify: check the degrees
Why: The first sum still has degree 3 and the second degree 2, matching the higher degree of each pair. A sum can drop in degree only if the leading terms cancel, which did not happen here — but it can, and noticing when it does is worth the glance.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 346-346
Trap
\[ (t^2 - 6t + 2) + (5t^2 - t - 8) \]
Add all the numbers that appear
Why: The coefficients are pooled without regard to which power they belong to.
\[ 6t^2 - 7t - 6 \;\to\; \text{written as } -7t^3 \quad \text{(wrong)} \]
A t squared term and a t term are different quantities and cannot be merged. At t equal to 2 the first is 4 and the second is 2 — not interchangeable.
\[ t^2 + 5t^2 = 6t^2; \quad -6t - t = -7t; \quad 2 - 8 = -6 \]
Combine only terms with the same variable and the same exponent
Why: Three separate additions, one per power, and the powers themselves never change.
\[ 6t^2 - 7t - 6 \]
Substituting a number settles any doubt about whether two terms are like: if they were, combining them could not change the value at any input.
Fill the middle
Example 1a, at the quadratic column.
Fill in the blanks
-5x^2 + 6x^2 = 1x^2
Why: Negative 5 plus 6 is 1, so the column gives x squared. A coefficient of 1 is written without the number, which is why the answer reads x squared rather than 1x squared — worth noting because a coefficient that vanishes to 1 is easy to lose entirely.
Comparison
Fill the blanks. Same rule, two layouts.
Comparison matrix
| Question | Vertical | Horizontal |
|---|---|---|
| How like terms are found | by aligning columns | by gathering them in a line |
| Missing powers | shown as an empty column | simply absent |
| Better for | long polynomials | short ones |
| The answer | the same | the same |
The second row is the practical difference: a vertical layout makes a missing power hard to overlook, which matters more the longer the polynomial gets.
Prediction
Commit before reasoning.
Predict first
You add a cubic and a quadratic. What degree is the result?
Correct: Always 3, the higher of the two.
\[ (2x^3 + 1) + (-2x^3 + 5) = 6: \; \text{degree drops from } 3 \text{ to } 0 \]
Why: The quadratic contributes nothing to the cubic term, so the cubic term survives untouched and the degree is 3. Cancellation can only happen when both polynomials have the same degree — adding 2x cubed and negative 2x cubed plus 5 gives a constant. So the honest rule is that the degree of a sum is at most the higher of the two, with equality unless the leading terms cancel.
Section
Section 2
Concept
Subtracting a polynomial means adding its opposite, and the opposite of a polynomial has every one of its signs flipped. Once that rewriting is done, the problem is an ordinary addition.
\[ -(3x^3 + 2x^2 - x + 7) = -3x^3 - 2x^2 + x - 7 \]
Write the flipped polynomial on its own line before combining anything. Doing the flip and the combining in one step is where the sign errors come from.
Figure (svg): A polynomial subtraction rewritten as adding the opposite, with every sign in the subtracted polynomial flipped
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 346-346 — Subtract polynomials vertically and horizontally
Picture it
Example 2a: the subtracted polynomial and its opposite.
Figure (svg): A polynomial subtraction rewritten as adding the opposite, with every sign in the subtracted polynomial flipped
All four signs flip, including the one that was already negative — negative x becomes positive x. That term is the one most often missed.
Worked example
Example 2a. Flip, then add.
\[ \text{Subtract } 3x^3 + 2x^2 - x + 7 \text{ from } 8x^3 - x^2 - 5x + 1. \]
Align like terms in columns
Why: Both polynomials are complete, so every column is filled.
Write the opposite of the subtracted polynomial
Why: Every sign flips: negative 3x cubed, negative 2x squared, positive x, negative 7.
\[ -3 x ^{3} - 2 x ^{2} + x - 7 \]
Add column by column
Why: Eight minus 3 is 5; negative 1 minus 2 is negative 3; negative 5 plus 1 is negative 4.
\[ 5 x ^{3} - 3 x ^{2} - 4 x \]
Add the constants
Why: One minus 7 is negative 6.
\[ -6 \]
Figure (svg): The solution to Worked example subtract vertically shown as a ladder of expressions, one row per algebraic move
\[ 5x^3 - 3x^2 - 4x - 6 \]
Verify: add the answer back to the subtracted polynomial
Why: Five x cubed minus 3x squared minus 4x minus 6, plus 3x cubed plus 2x squared minus x plus 7, gives 8x cubed minus x squared minus 5x plus 1 — the polynomial that was subtracted from. Addition undoes subtraction here exactly as it does for numbers, so this check is complete rather than partial.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 346-346
Fill the middle
Example 2b, at the flip.
Fill in the blanks
-(5z^2 - z + 3) = -5z^2 + z - 3
Why: The opposite of negative z is positive z. This is the term that gets missed, because its sign was already negative and it is easy to read a flip as making things negative rather than as reversing them. Every sign reverses, whichever way it was pointing.
Worked example
Example 2b and Guided Practice 2.
\[ \text{Compute } (4z^2 + 9z - 12) - (5z^2 - z + 3) \text{ and } (8d - 3 + 9d^3) - (d^3 - 13d^2 - 4). \]
First: flip every sign of the second bracket
Why: Negative 5z squared, positive z, negative 3.
\[ -5 z ^{2} + z - 3 \]
First: combine like terms
Why: Four minus 5 is negative 1; 9 plus 1 is 10; negative 12 minus 3 is negative 15.
\[ -z ^{2} + 10 z - 15 \]
Second: put the first polynomial in standard form
Why: Reordering gives 9d cubed plus 8d minus 3.
\[ 9 d ^{3} + 8 d - 3 \]
Second: flip and combine
Why: The opposite is negative d cubed plus 13d squared plus 4; combining gives 8d cubed, 13d squared, 8d and 1.
\[ 8 d ^{3} + 13 d ^{2} + 8 d + 1 \]
Figure (svg): The solution to Worked example two more differences shown as a ladder of expressions, one row per algebraic move
\[ -z^2 + 10z - 15, \qquad 8d^3 + 13d^2 + 8d + 1 \]
Verify: watch the term that appears from nowhere
Why: The second answer has a 13d squared term even though the first polynomial had none: it came entirely from flipping the negative 13d squared in the subtracted polynomial. A subtraction can introduce a power that neither original polynomial displayed positively, which is why reordering into standard form first is worth the line.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 346-346
Error analysis
A student subtracts one polynomial from another.
Annotate
On: \( (4z^2 + 9z - 12) - (5z^2 - z + 3) = 4z^2 + 9z - 12 - 5z^2 - z + 3 \)
The minus sign belongs to the whole bracket. Write the opposite polynomial on its own line before combining, and the error becomes impossible.
Matching
Flip every sign.
Match the pairs
Why: Every sign reverses, including ones that were already negative. In the third, two of the three terms become positive, which is a good illustration that the opposite of a polynomial is not a mostly negative one — it is a term-by-term reversal.
Ranking
Subtracting one polynomial from another.
Put in order
Why: Writing the opposite on its own line before combining is the step that prevents the standard error, and it costs one line. The check at the end is complete rather than partial: if the answer plus the subtracted polynomial returns the original, nothing can be wrong.
Prediction
Commit before reasoning.
Predict first
Does A minus B give the same answer as B minus A?
Correct: No — the two answers are opposites of each other.
\[ A - B = -(B - A) \]
Why: Subtracting the other way round flips every sign of the result, exactly as with numbers: 8 minus 3 is 5 and 3 minus 8 is negative 5. So reading the problem carefully matters — subtract A from B means B minus A, with the phrase reversing the order. Example 2a is worded that way deliberately, and getting it backwards gives an answer that is right in size and wrong in every sign.
Section
Section 3
Concept
To multiply two polynomials, multiply each term of the first by each term of the second and then combine like terms. FOIL is the special case where both have two terms; the general rule needs no new idea, only care that no pair is missed.
\[ (x+3)(3x^2 - 2x + 4) = 3x^3 + 7x^2 - 2x + 12 \]
Three factors are handled by multiplying two of them first and then multiplying the result by the third. Which two you start with does not matter.
Figure (svg): A multiplication grid pairing each term of a binomial with each term of a trinomial
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 347-347 — Multiply polynomials
Picture it
Example 3b: two terms times three terms.
Figure (svg): A multiplication grid pairing each term of a binomial with each term of a trinomial
Six cells for two terms times three, and the like terms are then collected down the diagonals. The grid is not required, but nothing else makes completeness so obvious.
Worked example
Example 3, both formats.
\[ \text{Multiply } -2y^2 + 3y - 6 \text{ by } y - 2, \text{ and } x + 3 \text{ by } 3x^2 - 2x + 4. \]
First, vertically: multiply by the constant
Why: Negative 2 times the trinomial gives 4y squared minus 6y plus 12.
\[ 4 y ^{2} - 6 y + 12 \]
First: multiply by the variable term
Why: Y times the trinomial gives negative 2y cubed plus 3y squared minus 6y, shifted one column left.
\[ -2 y ^{3} + 3 y ^{2} - 6 y \]
First: add the two rows
Why: Combining gives negative 2y cubed plus 7y squared minus 12y plus 12.
\[ -2 y ^{3} + 7 y ^{2} - 12 y + 12 \]
Second, horizontally: distribute the bracket over each term
Why: The quantity x plus 3 multiplies 3x squared, then negative 2x, then 4.
Second: combine like terms
Why: The squared terms 9x squared and negative 2x squared give 7x squared; the linear terms negative 6x and 4x give negative 2x.
\[ 3 x ^{3} + 7 x ^{2} - 2 x + 12 \]
Figure (svg): The solution to Worked example multiply two polynomials shown as a ladder of expressions, one row per algebraic move
\[ -2y^3 + 7y^2 - 12y + 12, \qquad 3x^3 + 7x^2 - 2x + 12 \]
Verify: check the degree and the constant
Why: Degree 2 times degree 1 gives degree 3 in both, as the exponents add. The constant term of each product is the product of the two constants: negative 6 times negative 2 is 12, and 3 times 4 is 12. Both checks pass, and each takes a second.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 347-347
Fill the middle
Example 3b, at the setup.
Fill in the blanks
(x+3)(3x^2 - 2x + 4): \; 2 \times 3 = 6 \text___
Why: Two terms in the first bracket and three in the second give six products before any combining. Counting first is the simplest guard against dropping a pair, and it generalises: three terms times four terms would give twelve.
Worked example
Example 4 and Guided Practice 4. Two at a time.
\[ \text{Multiply } (x-5)(x+1)(x+3) \text{ and } (a-5)(a+2)(a+6). \]
First: multiply the first two brackets
Why: The quantity x minus 5 times x plus 1 is x squared minus 4x minus 5.
\[ x ^{2} - 4 x - 5 \]
First: multiply that by the third
Why: Each of the three terms multiplies x and then 3, giving six products.
\[ x ^{3} - x ^{2} - 17 x - 15 \]
Second: multiply the first two
Why: A minus 5 times a plus 2 is a squared minus 3a minus 10.
\[ a ^{2} - 3 a - 10 \]
Second: multiply by a plus 6
Why: The six products combine to a cubed plus 3a squared minus 28a minus 60.
\[ a ^{3} + 3 a ^{2} - 28 a - 60 \]
Figure (svg): The solution to Worked example multiply three binomials shown as a ladder of expressions, one row per algebraic move
\[ x^3 - x^2 - 17x - 15, \qquad a^3 + 3a^2 - 28a - 60 \]
Verify: read the constant term and the zeros
Why: The constant term of the first is negative 5 times 1 times 3, or negative 15 — the product of the three constants, as expected. And because the factors vanish at 5, negative 1 and negative 3, those are the function's zeros, so the product is the intercept form of Lesson 4.2 raised to degree 3. Substituting x equal to 5 gives 125 minus 25 minus 85 minus 15, which is 0.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 347-347
Error analysis
A student multiplies a binomial by a trinomial.
Annotate
On: \( (x+3)(3x^2 - 2x + 4) = 3x^3 - 2x^2 + 4x + 9x^2 - 6x \)
Count the products before combining: the count is the number of terms in one bracket times the number in the other. A grid makes the count automatic.
Prediction
Commit before reasoning.
Predict first
You multiply a polynomial of degree 3 by one of degree 2. What degree is the product?
Correct: 5, the sum of the two.
\[ (a_3x^3)(b_2x^2) = a_3b_2x^5 \]
Why: The highest-power term of the product comes from multiplying the two leading terms, and their exponents add by the product of powers rule from Lesson 5.1. Since the leading coefficients are non-zero, their product is non-zero and the term survives, so the degree is exactly 5 with no exception — unlike a sum, where cancellation can lower the degree.
Ranking
Multiplying three binomials.
Put in order
Why: Combining after the first multiplication matters: it keeps the second multiplication down to three terms times two rather than four times two. Which pair you start with is free, since multiplication is associative, and choosing the pair that combines most is a small saving worth taking.
Comparison
Fill the blanks. One is a special case of the other.
Comparison matrix
| Question | FOIL | General rule |
|---|---|---|
| Applies to | two terms times two terms | any number of terms in each |
| Number of products | four | terms times terms |
| For (x+3)(3x^2-2x+4) | does not apply: three terms in the second | six products |
| The underlying property | distributive | distributive |
FOIL is a mnemonic for one case, not a separate method. Learning the general rule makes it unnecessary, and avoids being stuck the moment a bracket has three terms.
Section
Section 4
Concept
A sum times a difference gives a difference of squares. A binomial squared gives a trinomial with a middle term of twice the product. A binomial cubed gives four terms with coefficients 1, 3, 3, 1 and alternating signs when the binomial is a difference.
\[ (a+b)(a-b) = a^2 - b^2, \; (a\pm b)^2 = a^2 \pm 2ab + b^2, \; (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]
The first two are Lesson 4.3's patterns read forwards instead of backwards. There they were used to factor; here they are used to expand.
Figure (svg): The three special product patterns with a worked instance of each
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 347-347 — Special Product Patterns
Picture it
Each with a worked instance.
Figure (svg): The three special product patterns with a worked instance of each
The warning at the bottom is the important part: a binomial squared is never the sum of the squares, and a binomial cubed is never the sum of the cubes.
Worked example
Example 5, all three parts. The letters stand for whole expressions.
\[ \text{Expand } (3t+4)(3t-4), \; (8x-3)^2, \; (pq+5)^3. \]
First: sum and difference
Why: Here a is 3t and b is 4, so the answer is 3t squared minus 4 squared.
\[ 9 t ^{2} - 16 \]
Second: square of a binomial
Why: Here a is 8x and b is 3, and the middle term is twice 8x times 3.
\[ 64 x ^{2} - 48 x + 9 \]
Third: identify a and b
Why: Here a is pq and b is 5, and the pattern needs four terms.
\[ a = p q, b = 5 \]
Third: apply the cube pattern
Why: The terms are pq cubed, 3 times pq squared times 5, 3 times pq times 25, and 125.
\[ p ^{3} q ^{3} + 15 p ^{2} q ^{2} + 75 p q + 125 \]
Figure (svg): The solution to Worked example use the patterns shown as a ladder of expressions, one row per algebraic move
\[ 9t^2 - 16, \; 64x^2 - 48x + 9, \; p^3q^3 + 15p^2q^2 + 75pq + 125 \]
Verify: expand the second one directly
Why: The quantity 8x minus 3, times itself, gives 64x squared minus 24x minus 24x plus 9, which is 64x squared minus 48x plus 9. The pattern and the direct multiplication agree, as they must — the pattern is only a record of that multiplication done once in general.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 348-348
Matching
Identify a and b first.
Match the pairs
Why: In the last, a is 3z squared rather than 3z, so a squared is 9z to the fourth — a reminder that the letters in a pattern stand for whole expressions. Getting that wrong is the same error as taking the square root of 9x squared to be 9x in Lesson 4.4.
Worked example
Guided Practice 5. A difference, so the signs alternate.
\[ \text{Expand } (xy - 4)^3. \]
Identify a and b
Why: Here a is xy and b is 4, and the pattern for a difference has alternating signs.
\[ a = x y, b = 4 \]
Write the four terms with coefficients 1, 3, 3, 1
Why: The powers of a fall from 3 to 0 while the powers of b rise from 0 to 3.
\[ a ^{3} - 3 a ^{2} b + 3 a b ^{2} - b ^{3} \]
Substitute
Why: The terms are xy cubed, 3 times xy squared times 4, 3 times xy times 16, and 64.
Simplify each term
Why: Recall that xy cubed is x cubed y cubed by the power of a product rule.
\[ x ^{3} y ^{3} - 12 x ^{2} y ^{2} + 48 x y - 64 \]
Figure (svg): The solution to Worked example a cube with two variables shown as a ladder of expressions, one row per algebraic move
\[ x^3y^3 - 12x^2y^2 + 48xy - 64 \]
Verify: test with numbers
Why: At x equal to 1 and y equal to 1 the original is the quantity 1 minus 4, cubed, which is negative 27. The expansion gives 1 minus 12 plus 48 minus 64, which is also negative 27. A single numerical test catches almost every slip in a four-term expansion.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 348-348
Trap
\[ (8x - 3)^2 \]
Square each term
Why: The exponent is applied to the two terms separately.
\[ 64x^2 + 9 \quad \text{(wrong)} \]
At x equal to 1 the original is the quantity 8 minus 3, squared, which is 25. The wrong version gives 73.
\[ (8x-3)^2 = (8x)^2 - 2(8x)(3) + 3^2 = 64x^2 - 48x + 9 \]
Multiply the binomial by itself, or use the pattern
Why: Squaring means multiplying by itself, and that always produces two cross terms.
\[ (a \pm b)^2 \neq a^2 \pm b^2, \qquad (a \pm b)^3 \neq a^3 \pm b^3 \]
The book flags both of these in an Avoid Errors note. Lesson 5.1's properties distribute a power over a product or a quotient; nothing distributes it over a sum.
Fill the middle
Example 5b.
Fill in the blanks
(8x - 3)^2 = (8x)^2 - 2(8x)(3) + 3^2 = 64x^2 - 48x + 9
Why: Twice 8x times 3 is 48x, and the minus sign comes from the binomial being a difference. The middle term is exactly what distributing the exponent would lose, and it is the largest term in the expression at most values of x.
Sorting
Look at the shape before multiplying.
Sort into buckets
Sort each product.
Recognising a pattern saves the products but never changes the answer. Multiplying out would give the same thing, more slowly.
Prediction
Commit before reasoning.
Predict first
The cube of a binomial has coefficients 1, 3, 3, 1. What would the fourth power have?
Correct: 1, 4, 6, 4, 1.
\[ (a+b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4 \]
Why: The coefficients are the rows of Pascal's triangle, in which each entry is the sum of the two above it: 1, 1 then 1, 2, 1 then 1, 3, 3, 1 then 1, 4, 6, 4, 1. Squaring gave 1, 2, 1 and cubing gave 1, 3, 3, 1, so the pattern was already visible. Lesson 10.2's binomial theorem states this in general and explains why the entries count the ways of choosing terms.
Section
Section 5
Concept
When one quantity is the product of two others, multiplying their models gives a model for the product. The degree of the result is the sum of the two degrees, and the answer is rounded to the number of significant digits of the less precise input.
\[ T = W \cdot O \]
The rounding convention matters here. Multiplying a three-digit model by a three-digit model does not produce six digits of accuracy, so the result is rounded back to three.
Figure (svg): Two models multiplied to give a third, with the total oil output graphed over twenty years
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 348-348 — Use polynomial models
Picture it
Example 6: a quadratic times a linear, giving a cubic.
Figure (svg): Two models multiplied to give a third, with the total oil output graphed over twenty years
At t equal to 20, meaning the year 2000, the model gives about 5570 thousand barrels a day. Neither input model alone answers that question.
Worked example
Example 6. Multiply, round, then evaluate.
\[ \text{With } W = -0.575t^2 + 10.9t + 548 \text{ and } O = -0.249t + 15.4, \text{ find } T \text{ and } T(20). \]
Identify what the product means
Why: Wells times barrels per well gives total barrels, so the product is the model wanted.
\[ T = W \times O \]
Multiply the two polynomials
Why: A quadratic times a linear gives a cubic, with six products before combining.
\[ ^\circ 3 \]
Combine like terms
Why: The result is 0.143175t cubed minus 11.5691t squared plus 31.408t plus 8439.2.
Round to three significant digits
Why: The input models each carry three significant digits, so the product should too.
\[ T = 0.143 t ^{3} - 11.6 t ^{2} + 31.4 t + 8440 \]
Evaluate at t equal to 20
Why: The year 2000 is 20 years after 1980.
\[ \text{about } 5570\text{ thousand barrels} \]
Figure (svg): The solution to Worked example build a total from two models shown as a ladder of expressions, one row per algebraic move
\[ T(20) \approx 5570 \text{ thousand barrels} \]
Verify: check the two factors separately at t equal to 20
Why: W at 20 is negative 0.575 times 400, or negative 230, plus 218, plus 548, which is 536 thousand wells. O at 20 is negative 4.98 plus 15.4, which is about 10.4 barrels. Their product is about 5580 thousand — matching the cubic to within the rounding. Evaluating the factors is a good check because it avoids the cubic's arithmetic entirely.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 348-348
Fill the middle
Guided Practice 6, at t equal to zero.
Fill in the blanks
4010 \times 79.4 = 318394 \;\Longrightarrow\; \text___ T
Why: The constant term of a product is always the product of the two constant terms, because it is the only pair with no t in it. Rounded to three significant digits it becomes 318,000. Checking this one coefficient takes seconds and catches most arithmetic errors in a long multiplication.
Worked example
Guided Practice 6. Depth times cost per foot.
\[ \text{With } D = 109t + 4010 \text{ and } C = 0.542t^2 - 7.16t + 79.4, \text{ find } T. \]
Identify the product
Why: Feet times dollars per foot gives dollars, so total cost is depth times cost per foot.
\[ T = D \times C \]
Multiply 109t by the quadratic
Why: The terms are 59.078t cubed, negative 780.44t squared and 8654.6t.
Multiply 4010 by the quadratic
Why: The terms are 2173.42t squared, negative 28,711.6t and 318,394.
Combine and round
Why: The squared terms give about 1390t squared and the linear terms about negative 20,100t.
Figure (svg): The solution to Worked example total drilling cost shown as a ladder of expressions, one row per algebraic move
\[ T = 59.1t^3 + 1390t^2 - 20{,}100t + 318{,}000 \]
Verify: check the constant term
Why: At t equal to zero the models give a depth of 4010 feet and a cost of 79.40 dollars per foot, whose product is 318,394 dollars — matching the constant term before rounding. The constant term of a product is always the product of the two constants, which makes it the easiest coefficient to check.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 348-348
Error analysis
A student builds a model for total oil output.
Annotate
On: \( T = W + O = -0.575t^2 + 10.9t + 548 + (-0.249t + 15.4) \)
Check the units before choosing the operation. If the units of the two models do not cancel or combine into the units wanted, the operation is wrong.
Prediction
Commit before reasoning.
Predict first
Multiplying two three-digit models gives coefficients like 0.143175. Why not keep them?
Correct: The extra digits suggest precision the data do not have.
\[ 0.143175 \;\to\; 0.143 \quad \text{three significant digits, as in the inputs} \]
Why: The input coefficient 0.575 is known to three digits, so it might really be anything from 0.5745 to 0.5755. Multiplying propagates that uncertainty, and the sixth digit of the product is meaningless. Rounding to three significant digits is an honest statement of what is known, not a loss of information — and it is why the textbook makes it an explicit instruction.
Matching
The degrees add.
Match the pairs
Why: In every case the degrees add, including the last where a degree-0 factor leaves the degree unchanged. The third is the familiar one: two linear dimensions give a quadratic area, which is why every area problem in Chapter 4 was a quadratic.
Ranking
Building a product model.
Put in order
Why: The first step is the only one that is about the situation rather than the algebra, and skipping it is how a correct multiplication ends up answering the wrong question. The last step also has a context part: t counts years since 1980, so t equal to 20 must be translated back into the year 2000.
Comparison
Fill the blanks. Each has a different difficulty.
Comparison matrix
| Operation | The rule | What goes wrong |
|---|---|---|
| Adding | combine like terms | combining unlike terms |
| Subtracting | add the opposite of every term | flipping only the first sign |
| Multiplying | every term times every term | missing a pair |
| Degree of the result | at most the higher, for a sum | exactly the sum, for a product |
The last row is the practical difference: a product's degree is completely predictable, while a sum's can drop if the leading terms cancel.
Pattern
One routine per operation.
For three or more factors, multiply two at a time and simplify between steps, which keeps each multiplication small.
Check
Subtracting. Flip every sign.
Check your understanding
Compute (4z^2 + 9z - 12) - (5z^2 - z + 3).
Answer: A
Why: The opposite of the second polynomial is -5z^2 + z - 3, and combining gives -z^2 + 10z - 15.
Check
Multiplying. Count the products.
Check your understanding
Expand (x + 2)(3x^2 - x - 5).
Answer: A
Why: The six products give 3x^3 - x^2 - 5x + 6x^2 - 2x - 10, which combines to 3x^3 + 5x^2 - 7x - 10.
Check
A pattern. Identify a and b.
Check your understanding
Expand (8x - 3)^2.
Answer: A
Why: With a = 8x and b = 3, the middle term is -2(8x)(3) = -48x and the last is +9.
Real world
A rectangular garden measures x metres by x plus 4 metres. A path of uniform width 1 metre is laid around the outside, and then the whole thing is covered by a greenhouse whose height is x minus 1 metres.
Discussion prompt
Write polynomials for the path's area and for the greenhouse's volume, both in standard form, and say what degree each has and why.
Hint: The outer rectangle is 2 metres wider in each direction.
Answer:
\[ \text{path} = (x+2)(x+6) - x(x+4) = x^2 + 8x + 12 - x^2 - 4x = 4x + 12 \]
\[ \text{volume} = (x+2)(x+6)(x-1) = (x^2+8x+12)(x-1) = x^3 + 7x^2 + 4x - 12 \]
The path's area is linear, degree 1, and the greenhouse's volume is cubic, degree 3.
The path result is the surprising one. Two quadratics were subtracted and their leading terms cancelled, dropping the degree from 2 to 1 — which is exactly the case where a sum or difference loses degree. It also has a sensible meaning: the path is one metre wide, so its area grows in proportion to the perimeter rather than to the area, and perimeter is linear.
Commit first
Answer, then rate your confidence honestly.
Predict first
Is the quantity a plus b, cubed, equal to a cubed plus b cubed?
Correct: No — there are two middle terms.
\[ (2+3)^3 = 125 \neq 8 + 27 = 35 \]
Why: The expansion is a cubed plus 3a squared b plus 3ab squared plus b cubed, with four terms rather than two. A numerical test settles it instantly: 2 plus 3, cubed, is 125, while 8 plus 27 is 35. Exponents distribute over products and quotients only, as Lesson 5.1 established, and this is the same error as squaring a sum — worth being certain about, because it reappears in every chapter from here on.
Explain it
They know FOIL and have just been given a binomial times a trinomial.
Discussion prompt
In four sentences or fewer, explain how to multiply when FOIL does not apply, and how to be sure nothing is missed.
Hint: Talk about pairing terms.
Answer:
FOIL is just a name for pairing each of two terms with each of two other terms, which gives four products. When one bracket has three terms you do exactly the same thing: pair each term of the first with each term of the second, which gives six products.
To be sure nothing is missed, count first — the number of products is the number of terms in one bracket times the number in the other — and then check that you wrote that many before combining. Drawing a small grid makes the count automatic.
Exit ticket
Name the weakest spot before you close the deck.
Predict first
Which of these would you least want handed to you cold?
Correct: Whichever you picked is tonight's ten minutes, and each has a one-line fix.
Why: For subtraction, write the opposite polynomial on its own line before combining. For products, count the pairs before you start and check the count afterwards. For the cube, write the coefficients 1, 3, 3, 1 down first and fill in the powers around them. For models, check the units: if they do not combine into the units wanted, the operation is wrong. Do five of your chosen kind rather than twenty mixed ones.
Connect it up
Paper. Fifteen minutes.
Draw it
Choose two polynomials of your own, one cubic and one quadratic, and put them through everything. Top left: add them in both a vertical and a horizontal format, and confirm the two answers match. Top right: subtract the quadratic from the cubic, writing the opposite polynomial on its own line, and then check by adding your answer back to the quadratic. Bottom left: multiply them, using a grid so every pair is visible, and count the cells before combining. Bottom right: write out all three special product patterns with letters, then apply each one to an example of your own and verify it by direct multiplication. In a margin, write the degree of each of your three answers and say which one is completely predictable and which is not.
If your grid has a different number of cells than terms times terms, one row or column has been missed — recount before combining anything.
Recap
Five things, and the last one is what the first four were for.
| If you see | Then |
|---|---|
| Terms with the same power | Combine their coefficients |
| A minus before a bracket | Flip every sign inside |
| Two brackets to multiply | Form terms times terms products |
| Same terms, opposite middle signs | Difference of squares |
| A binomial squared or cubed | Use the pattern, never distribute |
| Two models with matching units | Multiply them |
This lesson expanded products into sums. Lesson 5.4 runs the process backwards, factoring polynomials of degree 3 and higher and using the result to solve equations.
McDougal Littell Algebra 2 (Texas Edition), Ch. 5 Polynomials and Polynomial Functions — Lesson 5.3 Add, Subtract, and Multiply Polynomials §5.3, pp. 346-351 — everything on these slides traces back here
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