Adding and subtracting polynomials. Includes monomials and the degree of a monomial, the vocabulary of polynomials, binomials and trinomials, writing a polynomial in standard form and naming it by degree and by number of terms, adding in vertical and horizontal formats, subtracting by adding the opposite, and modelling an area as a difference of polynomials.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 10 — Polynomials and Factoring
Adding and Subtracting Polynomials
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 568-574 — the lesson these objectives are drawn from
Warm-up
Chapter 2 combined like terms. This lesson does the same thing on longer expressions and gives the results names.
Discussion prompt
Simplify three x squared plus five x minus two x squared plus x. Which terms combined and which could not?
Hint: Only matching variable parts combine.
Answer:
\[ 3x^2 + 5x - 2x^2 + x = x^2 + 6x \]
The two squared terms combined into one and the two x terms combined into another, but a squared term and a plain x term never combine with each other. That is the whole arithmetic of this lesson; the rest is vocabulary and bookkeeping.
Concept
A monomial is a number, a variable, or a product of a number and one or more variables with whole number exponents. A polynomial is a monomial or a sum of monomials.
polynomial — A monomial or a sum of monomials. A polynomial of two terms is a binomial and one of three terms is a trinomial.
Adding and subtracting them is combining like terms.
Figure (svg): The degree of a monomial as the sum of its exponents
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 568-569
Section
Section 1
Concept
The degree of a monomial is the sum of the exponents of its variables. A number on its own has degree nought, because it can be written as that number times a variable to the power nought.
The exponents must be whole numbers for the expression to be a monomial.
Figure (svg): The degree of a monomial as the sum of its exponents
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 568-568 — the definition of a monomial and its degree, and Example 1
Picture it
One variable, two, or none.
Figure (svg): The degree of a monomial as the sum of its exponents
The two-variable case is the only one where any adding happens. Everything else is reading an exponent off the page or knowing that a constant hides an exponent of nought.
Worked example
This is Example 1 from the textbook.
\[ \text{State the degree of } -5x^4, \; 21b^3 \text{ and } 12. \]
Read the first exponent
Why: The coefficient plays no part.
\[ 4 \]
Read the second
Why: The variable is b rather than x.
\[ 3 \]
Rewrite the third
Why: Twelve is twelve times x to the nought.
\[ 12 x ^{0} \]
Read its exponent
Why: Nought.
\[ 0 \]
Figure (svg): The degree of a monomial as the sum of its exponents
\[ 4, \quad 3, \quad 0 \]
Verify: check that the coefficients were ignored
Why: The negative five and the twenty-one never entered the answers, because degree counts exponents rather than coefficients. A monomial like negative five x to the fourth has degree four just as x to the fourth does.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 568-568
Matching
Read the exponents.
Match the pairs
Why: Two of these have degree three by different routes: one from a single exponent and one from adding two. Degree does not record how it was arrived at, only the total.
Worked example
Guided Practice 1 to 4, with a two-variable case added.
\[ \text{State the degree of } 6x^3, \; -4p, \; 10, \; 3a^5 \text{ and } 3x^2y. \]
Take the first two
Why: Three, and an invisible one.
\[ 3, \; 1 \]
Take the constant
Why: Ten is ten times x to the nought.
\[ 0 \]
Take the fourth
Why: The exponent is five.
\[ 5 \]
Take the product
Why: Two plus one.
\[ 3 \]
Figure (svg): The degree of a monomial as the sum of its exponents
\[ 3, \; 1, \; 0, \; 5, \; 3 \]
Verify: check the invisible exponent
Why: A variable written without an exponent carries an exponent of one, so negative four p has degree one. That convention and the degree-nought convention for constants are the two places where an exponent is present without being written.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 568-568
Trap
\[ 21b^3 \;\Longrightarrow\; \text{degree } 21 \]
Take the largest number in the monomial as the degree
Why: Twenty-one is the biggest number visible.
Degree counts exponents, not coefficients. The exponent here is three, and the twenty-one describes how many of them there are rather than how many times b is multiplied by itself.
\[ 21b^3 \;\Longrightarrow\; \text{degree } 3 \]
Look only at what is written as an exponent
Why: Coefficients never affect degree.
This matters immediately, because degree is what decides the name and the standard-form ordering.
Faded example
A constant has one too.
Fill in the blanks
12 = 12x^0} \;\Longrightarrow\; \text0 ___
Why: Writing the constant with its invisible variable makes the degree obvious rather than arbitrary. It is the same convention that made zero exponents useful back in Lesson 8.2.
Elimination
Whole number exponents only.
Eliminate the wrong options
Which expression fails the definition?
Survives elimination: A
Why: Dividing by a variable is the same as an exponent of negative one, and the definition requires whole number exponents. Expressions with variables in a denominator are the subject of Chapter 11 and are deliberately excluded here.
Socratic
It is only a count of exponents.
Discussion prompt
Give two things the degree of a polynomial tells you. Then say why constants are given degree nought rather than being left undefined.
Hint: Think about graphs and about ordering.
Answer:
It fixes the standard-form ordering, since terms are arranged from largest exponent down, and it names the polynomial — degree two is quadratic, degree three is cubic — which tells you what kind of graph and how many solutions to expect. Chapter 9's whole discussion of two solutions and one turning point was really a statement about degree two.
Giving a constant degree nought keeps every rule uniform: it is genuinely that number times x to the nought, so the ordering rule places it last without needing an exception, and the naming rule calls it a constant polynomial rather than having to skip it. Leaving it undefined would mean special-casing constants in every statement about degree.
Section
Section 2
Concept
A polynomial is written in standard form when its terms are arranged from the largest exponent to the smallest. It is named by its degree and, separately, by its number of terms.
The degree of a polynomial in one variable is its largest exponent.
Figure (svg): Polynomials named by degree and by number of terms
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 569-569 — the definitions of binomial, trinomial and standard form, and Example 2
Picture it
Two independent labels.
Figure (svg): Polynomials named by degree and by number of terms
Nothing links the two columns: a binomial can have any degree and a quadratic can have two or three terms. Answering both parts separately is what the question usually asks for.
Worked example
This is Example 2 from the textbook.
\[ \text{Name } 6, \; 3x + 1, \; x^2 - 2x + 5 \text{ and } 4x^3 - 8x \text{ by degree and by terms.} \]
Take the constant
Why: Degree nought, one term.
Take the second
Why: Degree one, two terms.
Take the third
Why: Degree two, three terms.
Take the fourth
Why: Degree three, two terms.
Figure (svg): Polynomials named by degree and by number of terms
\[ \text{constant monomial}, \; \text{linear binomial}, \; \text{quadratic trinomial}, \; \text{cubic binomial} \]
Verify: notice that binomial appears twice
Why: The second and fourth are both binomials at different degrees, which shows the two names carry independent information. A question asking only for one of them is asking for half of what is available.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 569-569
Sorting
Exponents must decrease all the way.
Sort into buckets
Sort each polynomial by whether it is in standard form.
Two of the unsorted ones simply have the constant written first, which is easy to spot. The other hides a cubic term in the middle, which is the case that misleads people into reading the wrong degree.
Worked example
Guided Practice 5 to 8, one of which needs reordering.
\[ \text{Name } 8x, \; 10x - 5, \; x^2 + 4x + 4 \text{ and } 24 - x^3. \]
Take the first two
Why: Already in standard form.
Take the third
Why: Already in standard form.
Reorder the fourth
Why: The cubic term comes first.
\[ -x ^{3} + 24 \]
Name it
Why: Degree three, two terms.
Figure (svg): A polynomial rearranged into standard form
\[ \text{linear monomial}, \; \text{linear binomial}, \; \text{quadratic trinomial}, \; \text{cubic binomial} \]
Verify: check the sign carried across
Why: The x cubed term was written as being subtracted, so it becomes negative x cubed at the front rather than positive. Reordering never changes a term's sign, and losing that minus would change the polynomial entirely.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 569-569
Error analysis
The student was asked whether four x squared plus five x minus three x cubed plus six is in standard form.
Annotate
On: \( \begin{aligned} &4x^2 + 5x - 3x^3 + 6 \\ &\text{the exponents are } 2, 1, 3, 0 \\ &\text{so it is in standard form} \end{aligned} \)
The consequence is not cosmetic: reading the degree off an unsorted polynomial gives two rather than three, which would make it quadratic instead of cubic. Sorting first and reading the degree afterwards avoids the whole problem.
Faded example
Each term moves with its own sign.
Fill in the blanks
5x^3 - 2x + x^2 + 7 \;\to\; 5x^3 + x^2 - 2x + 7
Why: The minus sign belongs to the two x term and travels with it. Reordering is a rearrangement of a sum, so nothing about any individual term changes.
Translation
Degree first, then term count.
Match the pairs
Why: Each answer has two words because the question has two parts. Giving only one of them, however correct, answers half the question.
Socratic
The sum is the same either way.
Discussion prompt
Give two practical reasons for writing polynomials in standard form. Then say which operation in this lesson depends on it most.
Hint: Think about comparing and about columns.
Answer:
First, it makes two polynomials comparable at a glance: written the same way, matching terms sit in matching positions and a difference between them is visible immediately. Second, it puts the degree in the leading position, so the polynomial's most important feature can be read without scanning the whole expression.
Adding in vertical format depends on it most, because the columns only line up if both polynomials are sorted the same way. Example 3 begins by rewriting both expressions in standard form for exactly that reason, and skipping it produces columns that mix a squared term with an x term.
Section
Section 3
Concept
To add polynomials, combine like terms. A vertical format lines up like terms in columns, and a horizontal format groups them with brackets. Both give the same answer.
Write the answer in standard form.
Figure (svg): Two polynomials added in vertical format
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 569-569 — Example 3, Add Polynomials, in both formats
Picture it
Each column combines on its own.
Figure (svg): Two polynomials added in vertical format
An empty space in a column means a coefficient of nought, which is why the first polynomial's cubic term comes down unchanged. Leaving the gap rather than closing it keeps the alignment honest.
Worked example
This is Example 3, part a, from the textbook.
\[ \text{Find } (5x^3 + x^2 - 2x + 7) + (3x^2 - 4x + 7). \]
Write both in standard form
Why: Both already are.
Combine the squared terms
Why: One plus three.
\[ 4 x ^{2} \]
Combine the x terms
Why: Negative two minus four.
\[ -6 x \]
Combine the constants
Why: Seven plus seven.
\[ 14 \]
Figure (svg): Two polynomials added in vertical format
\[ 5x^3 + 4x^2 - 6x + 14 \]
Verify: substitute a value into both sides
Why: At x equal to one the two original polynomials give eleven and six, totalling seventeen, and the answer gives five plus four minus six plus fourteen, which is also seventeen. Substituting a small number checks a whole line of algebra in one step.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 569-569
Sorting
Same variable, same exponent.
Sort into buckets
Sort each pair by whether the two terms can be combined.
Two constants count as like terms, since both have degree nought. That is one more small payoff from giving constants a degree rather than leaving them outside the system.
Worked example
This is Example 3, part b.
\[ \text{Find } (2x^2 + x - 5) + (x + x^2 + 6). \]
Group the squared terms
Why: Two of them.
\[ (2 x ^{2} + x ^{2}) \]
Group the x terms
Why: Two of them.
\[ (x + x) \]
Group the constants
Why: Negative five and six.
\[ (-5 + 6) \]
Combine each group
Why: Coefficients only.
\[ 3 x ^{2} + 2 x + 1 \]
Figure (svg): Which terms can be combined and which cannot
\[ 3x^2 + 2x + 1 \]
Verify: check with a substitution
Why: At x equal to two the originals give five and twelve, totalling seventeen, and the answer gives twelve plus four plus one, which is seventeen. The second polynomial was not in standard form and the horizontal method handled it anyway, since grouping does not depend on the order.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 569-569
Trap
\[ 3x^2 + 2x = 5x^2 \]
Add the coefficients and keep the larger power
Why: Both terms have an x, so they looked combinable.
A squared term and a plain x term are not like terms and cannot be combined at all. Substituting x equal to two shows it: the left side is twelve plus four, which is sixteen, and the right side is twenty.
\[ 3x^2 + 2x \quad \text{is already simplified} \]
Combine only terms with identical variable parts
Why: Same variable and same exponent.
An answer with several terms left in it is usually finished rather than unfinished.
Faded example
Coefficients only.
Fill in the blanks
(5x^3 + x^2 - 2x + 7) + (3x^2 - 4x + 7) = 5x^3 + 4x^2 - 6x + 14
Why: The cubic term had no partner, so it comes down unchanged with a coefficient of five. Both x terms were negative, so their coefficients added to negative six rather than cancelling.
Elimination
Adding two trinomials.
Eliminate the wrong options
What is the sum of 2x squared + x - 5 and x squared + x + 6?
Survives elimination: A
Why: Each group of like terms combines separately and the variable parts are untouched. Option B is the persistent confusion between adding like terms and multiplying powers, which Lesson 8.1 governs and which does not apply here.
Socratic
Lesson 8.1 added exponents.
Discussion prompt
Explain why adding three x squared to two x squared leaves the exponent alone, when multiplying them would not. Then say what the terms are really counting.
Hint: Think of x squared as a single object.
Answer:
Adding is counting: three of a thing plus two of the same thing is five of that thing, and the thing itself does not change. Multiplying is different, because three x squared times two x squared means x squared appears twice more in a product, which is where Lesson 8.1's rule about adding exponents comes from.
The terms are counting copies of a variable part. Once you read three x squared as three copies of x squared, it becomes obvious that they can only be pooled with other copies of x squared, and that pooling five of them still leaves them as copies of x squared rather than of anything else.
Section
Section 4
Concept
To subtract one polynomial from another, add its opposite. Multiply every term of the subtracted polynomial by negative one, then add as usual.
\[ A - B = A + (-B) \]
Every term inside the brackets changes sign, not just the first.
Figure (svg): Subtraction rewritten as adding the opposite
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 570-570 — Example 4, Subtract Polynomials, and its Study Tip on changing signs
Picture it
Including the one on the constant.
Figure (svg): Subtraction rewritten as adding the opposite
The minus in front of the bracket distributes to everything inside it. Applying it to only the first term is the single most common error in the whole chapter.
Worked example
This is Example 4, part a, from the textbook.
\[ \text{Find } (2x^3 + 5x^2 - 4x + 8) - (2x^3 + 3x - 4). \]
Change every sign in the second
Why: Three terms flip.
\[ -2 x ^{3} - 3 x + 4 \]
Combine the cubic terms
Why: Two minus two.
\[ 0 \]
Combine the squared and x terms
Why: Nothing to pair with the squared; the x terms add.
\[ 5 x ^{2}, \; - 7 x \]
Combine the constants
Why: Eight plus four.
\[ 12 \]
Figure (svg): Subtraction rewritten as adding the opposite
\[ 5x^2 - 7x + 12 \]
Verify: substitute a value
Why: At x equal to one the originals give eleven and one, so the difference is ten, and the answer gives five minus seven plus twelve, which is also ten. The cubic terms cancelled completely, which is why the answer is a degree lower than either original.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 570-570
Faded example
The minus reaches all three terms.
Fill in the blanks
-(2x^2 - x - 4) = -2x^2 + x + 4
Why: Two of the three terms were negative and both become positive. Only the leading term was positive to start with, which is why copying the rest unchanged produces a wrong answer that still looks plausible.
Worked example
This is Example 4, part b.
\[ \text{Find } (3x^2 - 5x + 3) - (2x^2 - x - 4). \]
Distribute the minus sign
Why: All three signs flip.
\[ 3 x ^{2} - 5 x + 3 - 2 x ^{2} + x + 4 \]
Group the squared terms
Why: Three minus two.
\[ (3 x ^{2} - 2 x ^{2}) \]
Group the x terms
Why: Negative five plus one.
\[ (-5 x + x) \]
Group the constants
Why: Three plus four.
\[ (3 + 4) \]
Figure (svg): Subtraction rewritten as adding the opposite
\[ x^2 - 4x + 7 \]
Verify: check the two sign flips that matter
Why: The negative x became positive x and the negative four became positive four, both because of the minus in front of the bracket. Those two are the ones most often missed, and either would change the answer visibly.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 570-570
Trap
\[ (3x^2 - 5x + 3) - (2x^2 - x - 4) = 3x^2 - 5x + 3 - 2x^2 - x - 4 \]
Change the sign of the leading term and copy the rest
Why: The minus sign was read as attaching to the bracket's first term only.
This gives x squared minus six x minus one instead of x squared minus four x plus seven. Substituting x equal to nought exposes it: the true difference is three minus negative four, which is seven, not negative one.
\[ = 3x^2 - 5x + 3 - 2x^2 + x + 4 \]
Rewrite the whole second polynomial with every sign flipped, before combining anything
Why: Do the flipping as its own step.
Checking the constant term alone catches this almost every time.
Elimination
Subtracting a trinomial.
Eliminate the wrong options
What is 3x squared - 5x + 3 minus 2x squared - x - 4?
Survives elimination: A
Why: All three signs in the second polynomial must flip. Checking at x equal to nought distinguishes the options instantly, since only the constant term survives there and it must be seven.
Prediction
Subtracting two cubics whose leading terms match.
Predict first
What is the degree of the difference?
Correct: Less than three, because the cubic terms cancel.
\[ (2x^3 + 5x^2 - 4x + 8) - (2x^3 + 3x - 4) = 5x^2 - 7x + 12 \]
Why: Identical leading terms subtract to nought, so the cubic term vanishes and the degree drops to whatever the next surviving term is. In Example 4 the result was quadratic even though both originals were cubic, which is a genuine feature of subtraction rather than an accident. Addition can do this too, when the leading coefficients are opposites.
Socratic
Subtracting directly might seem simpler.
Discussion prompt
Explain the advantage of rewriting a subtraction as adding the opposite. Then say what makes the vertical format particularly prone to sign errors without that step.
Hint: How many rules do you then need?
Answer:
It reduces two operations to one: once the signs are flipped, every remaining step is addition, and there is only one rule to apply rather than two that must be kept straight column by column. It is the same reasoning that made subtracting integers easier in Chapter 2.
In vertical format the minus sign sits outside the second row where it is easy to lose sight of, and it must then be remembered separately for every single column. Flipping the signs once, in writing, converts that ongoing obligation into a single act that can be checked at a glance before any adding starts.
Section
Section 5
Concept
An area that is hard to compute directly can often be found as the difference of two rectangles. Writing each as a polynomial turns a geometry problem into a subtraction.
The verbal model is what keeps the order of subtraction right.
Figure (svg): A rectangular pool inside a larger rectangle, with the walkway between them
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 570-570 — Example 5, Subtracting Polynomials, and its verbal model for the walkway area
Picture it
The ring between two rectangles.
Figure (svg): A rectangular pool inside a larger rectangle, with the walkway between them
The walkway is a frame, which has no single length and width of its own. Subtracting the inner rectangle from the outer one is far easier than cutting the frame into four pieces.
Worked example
The lesson's pool situation, with dimensions stated.
\[ \text{A pool } 3x \text{ by } x \text{ sits inside a region } 6x \text{ by } (x + 6). \text{ Find the walkway's area.} \]
Write the verbal model
Why: The textbook's model.
\[ \text{walkway } =\text{ total } -\text{ pool} \]
Find the total area
Why: Six x times x plus six.
\[ 6 x ^{2} + 36 x \]
Find the pool's area
Why: Three x times x.
\[ 3 x ^{2} \]
Subtract
Why: Only the squared terms interact.
\[ 3 x ^{2} + 36 x \]
Figure (svg): A rectangular pool inside a larger rectangle, with the walkway between them
\[ (6x^2 + 36x) - 3x^2 = 3x^2 + 36x \]
Verify: test with a number
Why: Take x equal to two: the total region is twelve by eight, an area of ninety-six, and the pool is six by two, an area of twelve, leaving eighty-four. The formula gives three times four plus thirty-six times two, which is twelve plus seventy-two, also eighty-four.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 570-570
Faded example
The order comes from the verbal model.
Fill in the blanks
(6x^2 + 36x) - 3x^2 = 3x^2 + 36x
Why: Only the squared terms had partners, so the thirty-six x came through unchanged. That is the same as the empty column in the vertical format, and it is why the answer keeps a linear term.
Worked example
An area must not come out negative.
\[ \text{Does } 3x^2 + 36x \text{ make sense as an area for every } x? \]
Try a positive value
Why: x equal to two.
\[ 84 \]
Try a larger one
Why: x equal to ten.
\[ 660 \]
Try a negative value
Why: x equal to negative one.
\[ -33 \]
Interpret
Why: Lengths cannot be negative.
\[ x > 0 \]
Figure (svg): A rectangular pool inside a larger rectangle, with the walkway between them
\[ 3x^2 + 36x > 0 \text{ requires } x > 0 \]
Verify: check against the dimensions
Why: The pool's width is three x and its length is x, both of which need x positive to be lengths at all. The algebra would happily accept a negative value, and the situation would not — the same distinction that Lesson 9.2 drew about times.
Trap
\[ 3x^2 - (6x^2 + 36x) = -3x^2 - 36x \]
Subtract the total from the pool
Why: The pool was mentioned first in the diagram, so it went first.
This gives a negative area for every positive x. The verbal model says walkway equals total minus pool, and reversing the order reverses every sign.
\[ (6x^2 + 36x) - 3x^2 = 3x^2 + 36x \]
Write the verbal model before touching any algebra
Why: The words fix the order.
A negative answer to an area question is an immediate signal to check the order.
Hypothesis
It is a frame around the pool.
Predict first
What makes the difference method easier here?
Correct: The frame has no single length and width, but both rectangles do.
The four-piece method gives the same polynomial, which is worth checking once if you doubt it.
Why: Area of a rectangle needs one length and one width, and the walkway is a ring rather than a rectangle. It could be cut into four rectangular pieces and their areas added, which works but needs four correct sets of dimensions instead of two. Subtracting is fewer steps and fewer chances to go wrong, and it generalises to frames of any shape whose inside and outside are both easy.
Sorting
Something removed from something larger.
Sort into buckets
Sort each situation by whether a difference of polynomials fits it.
The three differences are all frames or leftovers, and the three sums are all combinations. Deciding which one a situation is, before writing anything, is what the verbal model step is for.
Socratic
The algebra could be started immediately.
Discussion prompt
Say what a verbal model protects you from. Then say how it would help if the problem had three regions rather than two.
Hint: Think about the order of the subtraction.
Answer:
It fixes the order and the operation before any expressions are written, which is exactly where the subtraction error creeps in. Writing walkway equals total minus pool in words makes it obvious which quantity is being taken away, and a diagram alone does not — the pool is drawn first and inside, which invites putting it first in the algebra.
With three regions the model becomes something like remaining equals total minus first minus second, and having those words down means the two subtractions are applied to the right thing in the right order. It also makes it visible whether the two removed pieces overlap, which is the case where a plain subtraction would double-count and the words force the question.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Adding | Subtracting | |
|---|---|---|
| First move | write both in standard form | flip every sign in the second polynomial |
| Then | combine like terms | combine like terms |
| Most common error | combining unlike terms | flipping only the first sign |
The middle row is identical in both columns, which is the point of rewriting subtraction as addition: after one extra step, there is only one procedure to run.
Pattern
To add or subtract any two polynomials, these five moves cover it.
Step two is worth doing as a separate written line rather than in your head, because it is the only step where a whole polynomial changes at once.
OpenStax Elementary Algebra 2e, §6.1 Add and Subtract Polynomials §6.1
Check
Degree counts exponents.
Check your understanding
What is the degree of the monomial 21b cubed?
Answer: A
Why: Degree is the sum of the exponents on the variables, and the only exponent here is three. The coefficient plays no part.
Check
Two names, two questions.
Check your understanding
How is 4x cubed - 8x classified?
Answer: A
Why: The largest exponent is three, making it cubic, and there are two terms, making it a binomial.
Check
Every sign flips.
Check your understanding
What is (3x squared - 5x + 3) minus (2x squared - x - 4)?
Answer: A
Why: All three terms of the second polynomial change sign, giving three x squared minus five x plus three minus two x squared plus x plus four.
Real world
This is the pool question from the lesson opener. A rectangular pool sits inside a larger rectangular region, and the walkway is what is left between them.
Discussion prompt
The whole region measures 6x by x + 6 and the pool measures 3x by x. Write a polynomial for the walkway's area, check it with a number, and say what values of x the model allows.
Hint: Walkway equals total minus pool.
Answer:
\[ (6x)(x + 6) - (3x)(x) = 6x^2 + 36x - 3x^2 = 3x^2 + 36x \]
Testing with x equal to two gives a total of ninety-six and a pool of twelve, leaving eighty-four — and the polynomial gives twelve plus seventy-two, also eighty-four.
Only positive values of x make sense, since three x and x are lengths of the pool and a length cannot be nought or negative. The polynomial itself is happy to be evaluated anywhere, and the situation is not, which is the same restriction that had to be applied by hand to the falling-object models in Chapter 9.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
Is 4x squared + 5x - 3x cubed + 6 written in standard form?
Correct: No, and its degree is 3.
\[ 4x^2 + 5x - 3x^3 + 6 = -3x^3 + 4x^2 + 5x + 6 \]
Why: Standard form requires the exponents to decrease from left to right, and here they run two, one, three, nought, which rises in the middle. Sorted properly it reads negative three x cubed plus four x squared plus five x plus six, and its degree is three — the largest exponent anywhere in the polynomial, not the exponent of whichever term happens to be written first. That is exactly why sorting matters: reading the degree off the unsorted version gives two and would misname a cubic polynomial as quadratic. Note also that the cubic term keeps its minus sign when it moves to the front.
Explain it
They wrote that three x squared minus five x plus three minus two x squared minus x minus four equals x squared minus six x minus one.
Discussion prompt
In no more than four sentences, explain which signs they missed and why. Then give them a check that would have caught it in one step.
Hint: Where does the minus in front of the bracket reach?
Answer:
A usable answer: the minus in front of the bracket multiplies every term inside it, not just the first. So negative x becomes positive x and negative four becomes positive four, giving x squared minus four x plus seven.
The check is to substitute nought for x, which kills every variable term and leaves only the constants. Three minus negative four is seven, so any answer whose constant is not seven is wrong, and that takes about five seconds.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Degree is fixed by looking only at exponents and remembering that a constant has degree nought. Naming is fixed by answering two separate questions rather than one. Subtraction signs are fixed by rewriting the whole second polynomial with flipped signs as its own line. Area models are fixed by writing the verbal model in words before any algebra. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page write four monomials — one with a single variable, one with two variables, one with a coefficient larger than its exponent, and one constant — and state each degree with the reasoning beside it. Underneath, build the naming table for four polynomials of degrees nought through three, filling in both the degree name and the term-count name, and mark clearly that the two columns are independent. In the middle, add two polynomials twice, once in vertical format with the columns ruled in and once horizontally with the like terms bracketed, and check both against a substitution of x equal to one. Beneath that, subtract two polynomials, writing the sign-flipped second polynomial as its own separate line before combining anything, and circle every sign that changed. In the lower corner, draw a rectangle inside a rectangle, label all four dimensions, write the verbal model in words, and find the area between them. Finally, in the margin, write the reason exponents stay unchanged when like terms are added.
Your sign-flipped line should have exactly as many changed signs as the second polynomial has terms. If fewer signs changed than there are terms, the minus did not reach all the way into the bracket.
Recap
Five things, and the third and fourth differ by exactly one step.
| If the question says | Your first move is |
|---|---|
| State the degree | Add the exponents; ignore coefficients |
| Write in standard form | Sort by exponent, carrying each sign |
| Identify the polynomial | Answer both by degree and by terms |
| Find the sum | Line up or group like terms |
| Find the difference | Flip every sign in the second, then add |
Lesson 10.2 multiplies polynomials instead of adding them, which brings the exponent rules of Chapter 8 back into play. That is where the distributive property has to reach every term of both factors rather than just one.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 10 Polynomials and Factoring — Lesson 10.1 Adding and Subtracting Polynomials §10.1, pp. 568-574 — everything on these slides traces back here
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