Polynomials and Factoring

Discussion prompt

Before we open Polynomials and Factoring: without looking back, what was the main idea of Real Numbers, Exponents, and Radicals, and what could you do by the end of it that you could not do before?

Hint: One sentence for the idea, one for the skill. If the second one is blank, that is the part to revisit.

Discussion prompt

A polynomial is a sum of terms, and every term is a number multiplied by a variable raised to a whole-number power.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Animation

Shows: Always take the GCF first — a rendered Manim animation.

Rendered with Manim.

Takeaway: Skipping this step makes every later step harder.

Sort into buckets

Every line below is part of the definition of term or of degree of a term — one or the other, never both. Put each where it belongs.

term

One piece of the polynomial, joined to the others by plus or minus.; Always read the sign in front of a term as part of that term.

degree of a term

The exponent on the variable in that term.; A plain number has degree zero.

b1

One piece of the polynomial, joined to the others by plus or minus. Always read the sign in front of a term as part of that term.

b2

The exponent on the variable in that term. A plain number has degree zero.

Step through it

Step through Reading one polynomial apart one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: term is 4x^3
2. Step 2: term is -7x^2
3. Step 3: term is x
4. Step 4: term is -9

Step through it

Step through Degree is the size of the biggest engine one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: x is 1
2. Step 2: x is 10
3. Step 3: x is 100

Animation

Shows: Factored form hands you the zeros — a rendered Manim animation.

Rendered with Manim.

Takeaway: The graph crosses exactly where a factor is zero.

Step through it

Step through The names you are expected to use one row at a time. What is driving the change, and what would the row after the last one be?

1. Step 1: degree is 0
2. Step 2: degree is 1
3. Step 3: degree is 2
4. Step 4: degree is 3
5. Step 5: degree is 4

Put in order

Put the moves of Worked example: describe a polynomial completely into the order they have to happen.

1. Order the terms from the largest exponent down
2. Read the degree off the first term
3. Read the leading coefficient off that same term
4. Verify by re-reading the original, unsorted version

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Standard form is a sorting job, nothing more.

Animation

Shows: Degree controls the shape — a rendered Manim animation.

Rendered with Manim.

Takeaway: A cubic can turn twice; a quadratic only once.

Match the pairs

Match each term to the definition this lesson gave it — not the one you would guess from the word.

• t1. term
• t2. coefficient
• t3. degree of a term
• t4. leading coefficient
• t5. like terms

• d1. One piece of the polynomial, joined to the others by plus or minus. Always read the sign in front of a term as part of that term.
• d2. The number multiplying the variable part of a term, sign included.
• d3. The exponent on the variable in that term. A plain number has degree zero.
• d4. The coefficient of the highest-degree term once the polynomial is in standard form. Sign included, always.
• d5. Terms with the same variable raised to the same exponent. Only the coefficients differ.

Why: These are the working definitions of term, coefficient, degree of a term, leading coefficient, like terms as Polynomials and Factoring uses them. Pairing them correctly is the test of whether you could state each one with the slide switched off.

Discussion prompt

Explain Think units, not letters by analogy to something with no College Algebra in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Discussion prompt

Worked example: adding two polynomials — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Drop the parentheses

Animation

Shows: Each line of the worked example "adding two polynomials", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The first polynomial becomes three minus five plus seven, which is five. The second becomes one plus eight minus two, which is seven. Their sum is twelve. The answer becomes four plus three plus five, which is also twelve, so the two expressions agree.

Discussion prompt

Explain Subtracting means adding the opposite to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Predict first

A student writes this, and it looks reasonable:

Change the sign on the first term only and copy the rest

It is wrong. Say what breaks — and say it before you turn the page.

Correct: The minus feels like it belongs to the term standing right next to it, so the six and the nine get copied down unchanged.

The minus feels like it belongs to the term standing right next to it, so the six and the nine get copied down unchanged.

Why: The minus feels like it belongs to the term standing right next to it, so the six and the nine get copied down unchanged.

Annotate

From Trap: the minus only reaches the first term — read this one piece at a time. What is each part doing?

On: \( \left(5x^{2} - 3x + 4\right) - \left(2x^{2} + 6x - 9\right) \)

• The minus feels like it belongs to the term standing right next to it, so the six and the nine get copied down unchanged.
• Subtracting a group means subtracting everything in it. Every sign inside flips, including the minus nine, which becomes plus nine.
• The correct answer plus the second polynomial rebuilds the first one exactly, so the correct answer is the true difference; the wrong version does not rebuild it.

Animation

Shows: Each line of the worked example "subtracting two polynomials", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: Three plus two gives five squared-terms; negative nine plus six gives negative three; thirteen minus nine gives four. That rebuilds the first polynomial exactly, so the subtraction was done correctly.

Eliminate the wrong options

Simplify (4x^2 - 7x + 1) - (x^2 - 3x - 6).

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. 3x^2 - 4x + 7
• B. 3x^2 - 10x - 5
• C. 3x^2 - 4x - 5
• D. 5x^2 - 10x - 5

Survives elimination: A

Why: Flip every sign in the second group to get 4x^2 - 7x + 1 - x^2 + 3x + 6. Then 4 - 1 = 3, and -7 + 3 = -4, and 1 + 6 = 7, giving 3x^2 - 4x + 7.

Check your understanding

Simplify (4x^2 - 7x + 1) - (x^2 - 3x - 6).

• A. 3x^2 - 4x + 7 (correct)
• B. 3x^2 - 10x - 5
• C. 3x^2 - 4x - 5
• D. 5x^2 - 10x - 5

Answer: A

Why: Flip every sign in the second group to get 4x^2 - 7x + 1 - x^2 + 3x + 6. Then 4 - 1 = 3, and -7 + 3 = -4, and 1 + 6 = 7, giving 3x^2 - 4x + 7.

Why B tempts people

The minus was never distributed at all: the second polynomial was copied down as written and then added, giving -7 - 3 = -10 and 1 - 6 = -5.

Why C tempts people

The sign flip stopped after the middle term. The -6 was copied instead of becoming +6, so the constant came out as 1 - 6 = -5.

Why D tempts people

The leading terms were added instead of subtracted, and the minus was never distributed, so both the 5x^2 and the rest are off.

Fill in the blanks

From Worked example: a monomial times a trinomial — finish the line. Write what belongs on the right of the equals sign before you look.

3x^12x^{5} \cdot 4x^___ = ___

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Three times four is twelve, and the exponents two and three add to five.

Animation

Shows: Each line of the worked example "a monomial times a trinomial", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The original becomes three times the quantity four minus two plus five, which is three times seven, or twenty-one. The answer becomes twelve minus six plus fifteen, which is also twenty-one.

Animation

Shows: Grouping, when there are four terms — a rendered Manim animation.

Rendered with Manim.

Takeaway: The shared bracket appearing is the signal that it worked.

Fill in the blanks

From Worked example: FOIL with a negative term — finish the line. Write what belongs on the right of the equals sign before you look.

2x \cdot 3x = 6x^{2}

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Two times three is six, and the exponents one and one add to two.

Animation

Shows: Each line of the worked example "FOIL with a negative term", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The original becomes nine times two, which is eighteen. The answer becomes twenty-four plus fourteen minus twenty, which is also eighteen.

Predict first

Two terms times three terms means six pairings. FOIL does not apply, so keep a layout that catches all six.

Commit before you compute: what does Worked example: a binomial times a trinomial come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify by substituting the value two

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. The original becomes five times the quantity eight minus ten plus four, which is five times two, or ten.

Animation

Shows: Each line of the worked example "a binomial times a trinomial", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The original becomes five times the quantity eight minus ten plus four, which is five times two, or ten. The answer becomes sixteen plus four minus twenty-two plus twelve, which is also ten.

Predict first

A student writes this, and it looks reasonable:

Squaring a binomial by squaring each piece:

It is wrong. Say what breaks — and say it before you turn the page.

Correct: The power of a product rule does let you square each factor of a product, and it is tempting to believe a sum behaves the same way.

A square is a product, so multiply it out:

Why: The power of a product rule does let you square each factor of a product, and it is tempting to believe a sum behaves the same way. It does not.

Draw it

Translate both ways. First write the expression above as a sentence with no symbols in it at all. Then cover it, and write your sentence back as notation. If the two versions disagree, the disagreement is the thing to fix.

Discussion prompt

Worked example: squaring a difference — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Identify the two pieces

Animation

Shows: Each line of the worked example "squaring a difference", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The original becomes two minus seven, squared, which is negative five squared, or twenty-five. The answer becomes four minus twenty-eight plus forty-nine, which is also twenty-five.

Discussion prompt

Pattern: multiplying any two polynomials works on the cases you have just seen. Push it to the edge: what is the most degenerate input it still handles — empty, zero, one item, everything equal — and what is the first case where it stops being true? Name the case, not just "it breaks".

Hint: Try the smallest legal input, then the largest, then the one where two things collide. Methods are specified at their edges; the middle takes care of itself.

Animation

Shows: Check by multiplying back — a rendered Manim animation.

Rendered with Manim.

Takeaway: Ten seconds of expanding catches almost every factoring error.

Check your understanding

Multiply and simplify (3x - 5)(2x + 7).

• A. 6x^2 - 35
• B. 6x^2 + 11x - 35 (correct)
• C. 6x^2 - 11x - 35
• D. 6x^2 + 11x + 35

Answer: B

Why: First gives 6x^2, outer gives 21x, inner gives -10x, and last gives -35. The middle terms combine as 21x - 10x = 11x, so the product is 6x^2 + 11x - 35.

Why A tempts people

Only the first and last pairings were done, as if the exponent distributed across each binomial. The outer and inner products, 21x and -10x, were never formed.

Why C tempts people

The signs on the outer and inner products were swapped, giving -21x + 10x = -11x instead of 21x - 10x = 11x.

Why D tempts people

The last product was taken as positive 35. Negative five times positive seven is negative thirty-five.

Check your understanding

Which expression equals (4x - 3)^2?

• A. 16x^2 + 9
• B. 16x^2 - 12x + 9
• C. 16x^2 - 24x + 9 (correct)
• D. 16x^2 - 24x - 9

Answer: C

Why: The square of a difference is the first piece squared, minus twice the product, plus the second piece squared: 16x^2 - 2(4x)(3) + 9 = 16x^2 - 24x + 9. Testing x = 1 gives 1 on both sides.

Why A tempts people

The exponent was distributed across the subtraction, squaring each piece and dropping the middle term entirely. Testing x = 1 gives 25 instead of 1.

Why B tempts people

The middle term was taken as one product of the pieces instead of twice that product, giving -12x where -24x belongs.

Why D tempts people

Negative three was squared as negative nine. Squaring a negative number gives a positive result, so the constant is positive nine.

Animation

Shows: Factoring is distributing, backwards — a rendered Manim animation.

Rendered with Manim.

Takeaway: The same identity, read in the other direction.

Sort into buckets

These are the pieces of Polynomials and Factoring, out of order. Put each one back under the part of the lesson it belongs to.

The Language of Polynomials

What a polynomial is; Term, coefficient, degree; Reading one polynomial apart

Multiplying Polynomials

One rule powers every multiplication; Worked example: a monomial times a trinomial; Multiplying is tiling a rectangle

Factoring: Multiplication Run Backwards

What factoring actually asks for; Sums hide information, products give it away; Step zero, always: the greatest common factor

s1

The Language of Polynomials is where Polynomials and Factoring puts What a polynomial is, Term, coefficient, degree, Reading one polynomial apart. Knowing which part of the lesson a problem belongs to is most of knowing which method to reach for.

s2

Multiplying Polynomials is where Polynomials and Factoring puts One rule powers every multiplication, Worked example: a monomial times a trinomial, Multiplying is tiling a rectangle. Knowing which part of the lesson a problem belongs to is most of knowing which method to reach for.

s3

Factoring: Multiplication Run Backwards is where Polynomials and Factoring puts What factoring actually asks for, Sums hide information, products give it away, Step zero, always: the greatest common factor. Knowing which part of the lesson a problem belongs to is most of knowing which method to reach for.

Predict first

The table runs: 12x^4 | 12 | x^4 · -18x^3 | -18 | x^3

In Worked example: factoring out the GCF, given the rows so far: what is the next one — the row where term is 30x^2?

Correct: 30x^2 | 30 | x^2

Why: The relationship between the columns, not the individual numbers, is what generates the next row. Twelve, eighteen, and thirty share a factor of six, and nothing larger works, since eighteen is not divisible by twelve.

Animation

Shows: Why factoring matters — a rendered Manim animation.

Rendered with Manim.

Takeaway: The zero-product property is the reason the whole chapter exists.

Predict first

A student writes this, and it looks reasonable:

Test it as a difference of squares immediately

It is wrong. Say what breaks — and say it before you turn the page.

Correct: Three is not a perfect square, so the first term is not a perfect square either, and twenty-seven is not a perfect square.

Three is not a perfect square, so the first term is not a perfect square either, and twenty-seven is not a perfect square. The pattern seems to fail.

Why: Three is not a perfect square, so the first term is not a perfect square either, and twenty-seven is not a perfect square. The pattern seems to fail.

Annotate

From Trap: calling it prime before pulling the GCF — read this one piece at a time. What is each part doing?

On: \( 3\left(x - 3\right)\left(x + 3\right) \)

• Three is not a perfect square, so the first term is not a perfect square either, and twenty-seven is not a perfect square. The pattern seems to fail.
• Substituting three gives twenty-seven minus twenty-seven, which is zero. A polynomial that is zero at three must have a factor built from three, so it cannot be prime.
• Both terms are divisible by three. Removing it is always the first move, and it often uncovers a pattern that was invisible before.

Comparison matrix

From Let the constant's sign do the thinking: refill the sign of middle column from what you know. The rest of the table is as it appeared.

Animation

Shows: Each line of the worked example "an easy trinomial", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: First gives the squared term, outer and inner give four and three of the first-power term for seven total, and last gives twelve. That rebuilds the original.

Put in order

Put the moves of Worked example: a trinomial with mixed signs into the order they have to happen.

1. Read the middle coefficient carefully
2. Read the signs
3. Search the factor pairs of twenty with opposite signs
4. Verify by multiplying back out

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. A bare minus in front of the variable means the coefficient is negative one, not zero.

Animation

Shows: When the leading coefficient is not one — a rendered Manim animation.

Rendered with Manim.

Takeaway: Splitting the middle term turns it into grouping.

Predict first

The table runs: -14 and 15 | -210 | 1 · -15 and 14 | -210 | -1

In Worked example: the ac method in full, given the rows so far: what is the next one — the row where pair is -10 and 21?

Correct: -10 and 21 | -210 | 11

Why: The relationship between the columns, not the individual numbers, is what generates the next row. Six times negative thirty-five is negative two hundred ten.

Animation

Shows: Each line of the worked example "the ac method in full", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: First gives six squared-units, outer gives negative ten of the first-power term, inner gives twenty-one of it, and last gives negative thirty-five. Twenty-one minus ten is eleven, matching the original.

Predict first

Worked example: the ac method again, faster is about to be worked. State your hypothesis first: which rule or definition decides this one, and what is the first move it forces? Then watch whether the example agrees with you.

Correct: Multiply four by negative fifteen

Why: The target product is negative sixty, and the target sum is negative four.

A hypothesis you wrote down is falsifiable; a vague sense of how it will go is not. If the example opens somewhere else, that gap is the thing worth chasing.

Animation

Shows: Each line of the worked example "the ac method again, faster", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: First gives four squared-units, outer gives six of the first-power term, inner gives negative ten of it, and last gives negative fifteen. Six minus ten is negative four, which matches.

Discussion prompt

Worked example: grouping, then keep going — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Split into the first two terms and the last two

Animation

Shows: Each line of the worked example "grouping, then keep going", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The original becomes one plus five minus four minus twenty, which is negative eighteen. The factored form becomes six times negative one times three, which is also negative eighteen.

Check your understanding

Factor 6x^2 - 7x - 20 completely.

• A. (2x - 5)(3x + 4) (correct)
• B. (2x + 5)(3x - 4)
• C. (6x - 5)(x + 4)
• D. (2x - 4)(3x + 5)

Answer: A

Why: Six times negative twenty is -120, and -15 plus 8 is -7, so split the middle term: 6x^2 - 15x + 8x - 20 = 3x(2x - 5) + 4(2x - 5) = (2x - 5)(3x + 4). Expanding gives 6x^2 + 8x - 15x - 20 = 6x^2 - 7x - 20.

Why B tempts people

The two signs were swapped. Expanding gives 6x^2 - 8x + 15x - 20 = 6x^2 + 7x - 20, which has the middle term with the wrong sign.

Why C tempts people

The numbers 5 and 4 were attached to the wrong factors. Expanding gives 6x^2 + 24x - 5x - 20 = 6x^2 + 19x - 20.

Why D tempts people

The first binomial still has a common factor of two, which means the greatest common factor was never removed. Expanding gives 6x^2 + 10x - 12x - 20 = 6x^2 - 2x - 20.

Animation

Shows: The difference of squares — a rendered Manim animation.

Rendered with Manim.

Takeaway: A sum of squares does not factor over the reals — only the difference.

Predict first

Now a case where the pattern applies twice:

Commit before you compute: what does Worked example: two differences of squares come out to? A rough magnitude and the right form is enough — the point is to have something concrete to be wrong about.

Correct: Verify by substituting the value one

Why: A prediction you can defend turns the computation into a check rather than a leap of faith — and an answer that contradicts it is caught on the spot. The original becomes sixteen minus eighty-one, which is negative sixty-five.

Animation

Shows: Each line of the worked example "two differences of squares", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The original becomes sixteen minus eighty-one, which is negative sixty-five. The factored form becomes negative one times five times thirteen, which is also negative sixty-five.

Predict first

A student writes this, and it looks reasonable:

Take the square root of each piece and use two plus signs

It is wrong. Say what breaks — and say it before you turn the page.

Correct: Both terms really are perfect squares, so the pattern looks like it should apply with the signs adjusted.

Both terms really are perfect squares, so the pattern looks like it should apply with the signs adjusted.

Why: Both terms really are perfect squares, so the pattern looks like it should apply with the signs adjusted.

Sort into buckets

Some of these hold up and some are the exact mistakes this lesson is built to prevent. Sort them.

Holds up

A polynomial is a sum of terms, and every term is a number multiplied by a variable raised to a whole-number power.; The degree of the polynomial is the largest degree among its terms.; Notice two habits worth building: a bare variable has an invisible coefficient of one, and the minus signs belong to the terms that follow them.

Breaks

Change the sign on the first term only and copy the rest; Squaring a binomial by squaring each piece:

sound

These are stated as this lesson states them — each one survives the edge cases Polynomials and Factoring puts it through.

flawed

Each of these is lifted from a trap in this deck: reasonable-sounding, and wrong in a way that only shows up once you rely on it.

Fill in the blanks

From Worked example: spotting a perfect square trinomial — finish the line. Write what belongs on the right of the equals sign before you look.

2 \cdot 3x \cdot 5 = 30x

Why: Producing the right-hand side unprompted is the difference between recognising this line and being able to use it. Nine with an even exponent is the square of three times the variable, and twenty-five is the square of five.

Animation

Shows: Recognising a perfect square — a rendered Manim animation.

Rendered with Manim.

Takeaway: Check the middle term against twice the root of the last.

Discussion prompt

Explain Sum and difference of cubes to a student a year behind you. No notation, no jargon they have not met — and it still has to be true.

Hint: If your explanation needs a symbol they have never seen, you are describing the notation rather than the idea.

Animation

Shows: Cubes factor too — a rendered Manim animation.

Rendered with Manim.

Takeaway: Same signs first, opposite sign in the middle of the quadratic.

Animation

Shows: Each line of the worked example "a difference of cubes", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The original becomes eight minus twenty-seven, which is negative nineteen. The factored form becomes negative one times the quantity four plus six plus nine, which is negative one times nineteen, or negative nineteen.

Animation

Shows: Each line of the worked example "a sum of cubes", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The original becomes twenty-seven plus sixty-four, which is ninety-one. The factored form becomes seven times the quantity nine minus twelve plus sixteen, which is seven times thirteen, or ninety-one.

Discussion prompt

Explain Some polynomials really are prime by analogy to something with no College Algebra in it at all — a queue, a recipe, a map, a bank balance, whatever fits. Then say where your analogy breaks.

Hint: An analogy that never breaks is not an analogy, it is the same idea wearing a hat. Find the seam — that is the part that is actually new.

Discussion prompt

Some higher-degree polynomials are quadratics in disguise. The tell is that one exponent is exactly double the other.

That is stated as though it always holds. Do one of two things: produce a case where it fails, or say precisely what rules such a case out. "It just does" is not on the menu.

Hint: Hunt at the extremes first — zero, one, negative, empty, equal. If every extreme survives, the reason they survive is the proof.

Discussion prompt

Worked example: factoring in quadratic form — before any calculation: what is the plan? Name the moves in order, in plain English, without doing the arithmetic.

Hint: It starts with: Substitute a single letter for the squared variable

Animation

Shows: Factoring a simple trinomial — a rendered Manim animation.

Rendered with Manim.

Takeaway: Product and sum — that pair of conditions is the whole search.

Put in order

Put the moves of Worked example: running the whole flowchart into the order they have to happen.

1. Step one: pull the greatest common factor
2. Step two: count the terms inside, which is two
3. Step three: factor every factor again
4. Verify by substituting the value one

Why: These are the moves of the worked example in the order it makes them, and each one is set up by the one before it. Both terms are divisible by two.

Animation

Shows: Each line of the worked example "running the whole flowchart", appearing one at a time.

The same working the example does, in the order a tutor would write it.

Takeaway: The original becomes two minus thirty-two, which is negative thirty. The factored form becomes two times negative one times three times five, which is also negative thirty.

Check your understanding

Factor 5x^3 - 45x completely.

• A. 5x(x^2 - 9)
• B. x(5x^2 - 45)
• C. 5x(x - 3)(x + 3) (correct)
• D. 5x(x - 3)^2

Answer: C

Why: The greatest common factor is 5x, leaving x^2 - 9, which is a difference of squares and splits into (x - 3)(x + 3). Multiplying back: 5x(x^2 - 9) = 5x^3 - 45x.

Why A tempts people

Correct so far but stopped one step early. The leftover x^2 - 9 is a difference of squares, so it still factors.

Why B tempts people

Only the variable was pulled out; the common factor of 5 in both coefficients was left behind, so the greatest common factor is incomplete.

Why D tempts people

This treats x^2 - 9 as a perfect square trinomial. Expanding gives 5x(x^2 - 6x + 9) = 5x^3 - 30x^2 + 45x, which is not the original.

Eliminate the wrong options

Which polynomial is prime over the integers?

3 of these 4 are wrong. Strike them one at a time, and say what rules each one out before you strike the next. The survivor is the answer.

• A. x^2 - 9
• B. x^2 + 9
• C. x^2 + 6x + 9
• D. 2x^2 + 8

Survives elimination: B

Why: A sum of squares with no common factor does not factor over the integers: no pair of integers multiplies to 9 and adds to 0. The other three all factor, as (x - 3)(x + 3), (x + 3)^2, and 2(x^2 + 4).

Check your understanding

Which polynomial is prime over the integers?

• A. x^2 - 9
• B. x^2 + 9 (correct)
• C. x^2 + 6x + 9
• D. 2x^2 + 8

Answer: B

Why: A sum of squares with no common factor does not factor over the integers: no pair of integers multiplies to 9 and adds to 0. The other three all factor, as (x - 3)(x + 3), (x + 3)^2, and 2(x^2 + 4).

Why A tempts people

This is a difference of squares, the most common factoring pattern in the course. It splits into (x - 3)(x + 3).

Why C tempts people

This is a perfect square trinomial: the outer terms are squares and the middle term is twice the product of x and 3, so it is (x + 3)^2.

Why D tempts people

Both terms share a factor of 2, so the greatest common factor comes out first and gives 2(x^2 + 4). Having a factorable greatest common factor means it is not prime.

Draw it

One page, no notation unless you need it: draw how these connect — The Language of Polynomials · Multiplying Polynomials · Factoring: Multiplication Run Backwards · The Special Forms · A Strategy You Can Trust. Put an arrow wherever one of them is what makes another possible, and label the arrow with why.