Week 6 · Chapter

Chapter 6 — Polynomials & Factoring

College Algebra Free sample edition

The module's primary reading · ~40–50 minutes · Objective 5 (add, subtract, and multiply polynomials, apply the special products, and factor polynomial expressions by the greatest common factor, trinomial patterns, the difference of squares, and grouping) · Read it first this week; the lecture, the tutorial, the quiz, and the Model/Build Studio all build on it.

In plain language

This week is about expressions with several pieces, like the area of a garden bed that has grown in two directions. You will learn to build these expressions by multiplying, and you will learn to take them apart again. Taking an expression apart into pieces that multiply is called factoring (rewriting a sum as a product, the way 12 can be rewritten as 3 times 4). Every answer this week has a built-in check: multiply your pieces back together and see whether you get what you started with.

Why you already care about this

Picture a square garden bed behind a community center. Each side is x feet long. The volunteers decide to make it bigger: 3 feet longer in one direction and 4 feet longer in the other. Now the bed is x + 3 feet by x + 4 feet, and its area is (x + 3)(x + 4) square feet. If the original side was 2 feet, the new bed is 5 feet by 6 feet, or 30 square feet of soil to order.

Multiplying the two sides out gives x² + 7x + 12. That expression is a polynomial, and it describes the area for any starting size at once. The same kind of expression gives the area of a framed poster, the volume of a box folded from a flat sheet, and the height of a ball thrown into the air. Those are four different settings, and one set of skills covers all of them.

This chapter teaches two moves that undo each other. You will multiply to build a polynomial, and you will factor to take it apart. You will also learn to check every answer yourself, so you never have to guess whether you are right.

By the end of this chapter you will be able to:

name the terms, degree, leading coefficient, and constant term of a polynomial;
add and subtract polynomials, sending a minus sign to every term it covers;
multiply polynomials with FOIL and check the product at a number;
use the three special products without dropping a middle term;
factor out a greatest common factor, and factor trinomials with and without a leading coefficient;
factor a difference of squares, explain why a sum of squares will not factor, and factor four terms by grouping.

Carry this sentence with you: Expand to check — every factoring, every time.

Where we are

Already covered: Week 1 gave you the exponent rules and the distributive property; Week 2 solved equations; Week 3 evaluated functions; Weeks 4 and 5 worked with lines and systems of lines.
This week: polynomials as objects — their vocabulary, how to add, subtract, and multiply them, and how to factor them.
How they connect: multiplying polynomials is the distributive property used many times, and every exponent you combine follows the Week 1 product rule.
Where we're going next: Week 7, Quadratic Equations, uses this week's factoring to solve equations such as x² − 6x + 5 = 0, which begins a new unit on curves.

Each earlier week hands you a tool you will lean on. Week 1 taught the distributive property, a(b + c) = ab + ac, which is the engine behind every product this week. It also taught the product rule for exponents, so you know that x · x = x² and 3x² · 2x = 6x³. Week 1 also defined like terms, the terms with the same variable raised to the same power, such as −6x and 2x. Week 3 taught you to evaluate an expression at a number, and this week you use that skill as your check. Week 4 met 3x + 4 as a line; this week x² terms join the picture.

1. Name the parts of a polynomial

Think about a grocery receipt. It is a list of items, each with an amount, added up. A polynomial works the same way. It is a list of pieces added together, and each piece is a number times a power of x.

A polynomial is an expression built by adding terms, where each term is a number times a variable raised to a whole-number exponent, such as 3x² − 2x + 5.
A term is one piece of a sum, separated from the next piece by a plus or minus sign; in 3x² − 2x + 5 the terms are 3x², −2x, and 5.
The degree is the largest exponent on the variable in a polynomial; the degree of 3x² − 2x + 5 is 2.
The leading coefficient is the number that multiplies the highest-power term of a polynomial; in 3x² − 2x + 5 the leading coefficient is 3.
The constant term is the term of a polynomial with no variable in it, the plain number at the end when the terms are written from highest power to lowest.

Polynomials also get names by how many terms they have.

A monomial is a polynomial with exactly one term, such as 3x² or 7.
A binomial is a polynomial with exactly two terms, such as x + 3 or 6x³ − 9x².
A trinomial is a polynomial with exactly three terms, such as x² + 7x + 12.

Worked example — naming a polynomial you already know. You evaluated 3x² − 2x + 5 back in Week 1. Now read it as a polynomial.

Count the terms. The pieces are 3x², −2x, and 5. That is three terms, so it is a trinomial.
Find the degree. The exponents are 2 on x², 1 on x, and none on the 5. The largest is 2, so the degree is 2.
Find the leading coefficient. The highest-power term is 3x², and the number in front is 3.
Find the constant term. The piece with no x is 5.

In words: 3x² − 2x + 5 is a degree-2 trinomial with leading coefficient 3 and constant term 5. Two more quick reads: 6x³ − 9x² is a binomial of degree 3 with leading coefficient 6, and 3x² is a monomial of degree 2.

Misconception check. Students often think the leading coefficient is whatever number comes first on the page. In 5 + 4x − x², the first number is 5, but the highest power is x², and its coefficient is −1. The cure: find the biggest exponent first, then read the number glued to that term, sign included. Writing the terms from highest power to lowest makes the leading coefficient easy to spot.

2. Add and subtract: send the minus to every term

Imagine you combine two fruit orders. One has 2 crates of apples and 3 bags of limes. The other has 4 crates of apples and 1 bag of limes. Together you have 6 crates of apples and 4 bags of limes. You added apples to apples and limes to limes. You never added a crate to a bag.

Polynomials add the same way. You combine like terms: x² terms with x² terms, x terms with x terms, and plain numbers with plain numbers.

Worked example — adding. Add (x² − 6x + 5) + (x² − 2x − 15).

Group like terms. (x² + x²) + (−6x − 2x) + (5 − 15).
Combine the x² terms. x² + x² = 2x².
Combine the x terms. −6x − 2x = −8x.
Combine the numbers. 5 − 15 = −10.

In words: the sum is 2x² − 8x − 10.

Subtracting needs one extra move. A minus sign in front of parentheses multiplies every term inside by −1. That is the distributive property from Week 1, with −1 as the number out front.

Worked example — subtracting. Find (x² − 6x + 5) − (x² − 2x − 15).

Send the minus to every term of the second polynomial. −(x²) = −x², −(−2x) = +2x, and −(−15) = +15. So the problem becomes x² − 6x + 5 − x² + 2x + 15.
Group like terms. (x² − x²) + (−6x + 2x) + (5 + 15).
Combine. x² − x² = 0, −6x + 2x = −4x, and 5 + 15 = 20.
Write the result. 0 − 4x + 20 = −4x + 20.

In words: the difference is −4x + 20. The x² terms cancel, so the difference is a binomial of degree 1.

Misconception check. Students often apply the minus only to the first term of the second polynomial. That gives x² − 6x + 5 − x² − 2x − 15 = −8x − 10, which is wrong. It is tempting because the minus sign sits right next to x² and looks like it belongs only there. The cure: before you combine anything, rewrite the second polynomial with every sign flipped. A check at x = 2 settles it: the original difference is (4 − 12 + 5) − (4 − 4 − 15) = −3 − (−15) = 12, and −4(2) + 20 = 12, while −8(2) − 10 = −26.

3. Multiply with FOIL and check at a value

Go back to the garden bed. Its area is (x + 3)(x + 4). To turn that product into a sum, every term of the first factor has to multiply every term of the second factor. That is the distributive property, used twice.

For two binomials there are exactly four products to make. A memory aid keeps them straight.

FOIL is a memory aid for multiplying two binomials: multiply the First terms, the Outer terms, the Inner terms, and the Last terms, then add the four products.

Worked example — the shared polynomial. Multiply (x − 1)(x − 5).

First. x · x = x².
Outer. x · (−5) = −5x.
Inner. (−1) · x = −x.
Last. (−1) · (−5) = 5. A negative times a negative is positive.
Add the four products. x² − 5x − x + 5.
Combine the like terms. −5x − x = −6x, so the product is x² − 6x + 5.

In words: (x − 1)(x − 5) and x² − 6x + 5 are two ways to write the same polynomial.

How can you be sure? Pick a number for x and evaluate both forms. They must agree. At x = 2, the factored form gives (2 − 1)(2 − 5) = (1)(−3) = −3, and the expanded form gives 2² − 6(2) + 5 = 4 − 12 + 5 = −3. They agree, so you can trust the product.

The two forms agree at every input, not only at 2. Here are seven inputs side by side.

Values of x² − 6x + 5 and of (x − 1)(x − 5) for x = 0 through 6
x	x² − 6x + 5	(x − 1)(x − 5)
0	5	5
1	0	0
2	−3	−3
3	−4	−4
4	−3	−3
5	0	0
6	5	5

The seven values are 5, 0, −3, −4, −3, 0, 5, and they add up to 0. Notice the zeros at x = 1 and x = 5. They show up exactly where one factor is zero. Keep that in mind for next week.

Worked example — the garden bed. Multiply (x + 3)(x + 4).

First, Outer, Inner, Last. x · x = x², x · 4 = 4x, 3 · x = 3x, and 3 · 4 = 12.
Add and combine. x² + 4x + 3x + 12 = x² + 7x + 12.
Check at x = 2. The factored form gives (5)(6) = 30, and the expanded form gives 4 + 14 + 12 = 30.

In words: a bed that started 2 feet on a side needs 30 square feet of soil, and the formula x² + 7x + 12 square feet works for any starting side. One more product in the same style: (x + 3)(x − 5) = x² − 5x + 3x − 15 = x² − 2x − 15.

In practice (your tools). Open the free Desmos graphing calculator. On line 1 type y = (x - 1)(x - 5), and on line 2 type y = x^2 - 6x + 5. The two curves land exactly on top of each other, which is Desmos's way of saying the expressions are equal. Now change line 2 to a wrong product, such as y = x^2 - 4x + 5, and watch the two curves split apart.

Misconception check. Students often multiply only the first terms and the last terms, writing (x + 3)(x + 4) as x² + 12. It is tempting because it looks like the way you multiply two plain numbers. The cure: count the products. Two terms times two terms is four products, so FOIL always makes four, and the outer and inner products are usually the middle term.

4. Use the three special products

Some products come up so often that it pays to know them on sight. Here is a campus example. A square poster is x inches on a side, and it goes into a frame that adds 3 inches on every side. The framed poster is x + 6 inches on a side, so its area is (x + 6)² square inches.

Picture the framed square split into four rectangles: the original x-by-x poster, two strips that are x inches by 6 inches, and a small 6-by-6 corner square. Those four pieces are x², 6x, 6x, and 36. The two strips are the part people forget.

The special products are three multiplication patterns worth knowing by sight: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b².
A perfect-square trinomial is a trinomial that is the square of a binomial, such as x² + 12x + 36 = (x + 6)²; its middle term is twice the product of the binomial's two terms.

Worked example — the square of a sum. Expand (x + 6)².

Name the parts. The first term is a = x, and the second is b = 6.
Use the pattern. a² + 2ab + b² = x² + 2(x)(6) + 6².
Simplify. 2(x)(6) = 12x, and 6² = 36, so (x + 6)² = x² + 12x + 36.
Check at x = 2. The left side is 8² = 64, and the right side is 4 + 24 + 36 = 64.

In words: the framed poster's area is x² + 12x + 36 square inches. The 12x is the two side strips.

Worked example — the square of a difference. Expand (3x − 2)².

Name the parts. a = 3x and b = 2.
Use the pattern. a² − 2ab + b² = (3x)² − 2(3x)(2) + 2².
Simplify. (3x)² = 9x², 2(3x)(2) = 12x, and 2² = 4.

In words: (3x − 2)² = 9x² − 12x + 4. The square of 3x is 9x², because the 3 gets squared too.

Worked example — a sum times a difference. A square patio is x feet on a side. A redesign makes it 4 feet longer and 4 feet narrower. Find its new area, (x + 4)(x − 4).

FOIL. x · x = x², x · (−4) = −4x, 4 · x = 4x, and 4 · (−4) = −16.
Combine. x² − 4x + 4x − 16. The two middle terms cancel, since −4x + 4x = 0.
Write the result. (x + 4)(x − 4) = x² − 16.

In words: the redesigned patio has 16 square feet less area than the original x² square feet, no matter how big the patio was. The extra length does not make up for the lost width.

Misconception check. Students often write (x + 6)² = x² + 36. It is tempting because squaring each piece feels like the natural move. But a square means the binomial times itself, and that makes four products, not two. A check at x = 2 shows the gap: the true value is 64, and x² + 36 gives 4 + 36 = 40. The cure: write (x + 6)(x + 6) and FOIL it once, or remember the poster's two side strips. Keep this one handy: (a + b)² has three terms, not two.

5. Factor out the greatest common factor first

Factoring runs multiplication backward. Picture three friends at a café. Each one orders a 6-dollar salad and a 9-dollar entrée. The total is 3 · 6 + 3 · 9, and it is also 3 · (6 + 9) = 3 · 15 = 45 dollars. Pulling out the shared 3 is factoring.

Factoring is rewriting a polynomial as a product of simpler polynomials; it undoes multiplication, and you check it by multiplying the factors back out.
The greatest common factor is the largest expression that divides every term of a polynomial, often written GCF; for 6x³ − 9x² it is 3x², so 6x³ − 9x² = 3x²(2x − 3).

Worked example — a GCF with a variable. Factor 6x³ − 9x².

Find the largest number that divides both coefficients. 6 and 9 share 3.
Find the largest power of x in both terms. x³ and x² share x².
Write each term as GCF times something. 6x³ = 3x² · 2x, and 9x² = 3x² · 3.
Pull out the GCF. 6x³ − 9x² = 3x²(2x − 3).
Check by expanding. 3x² · 2x − 3x² · 3 = 6x³ − 9x².

In words: 6x³ − 9x² factors as 3x² times (2x − 3).

Worked example — the launched ball. A ball launched upward from an 80-foot platform at 64 feet per second has height −16t² + 64t + 80 feet after t seconds. Factor out the GCF.

Find the GCF. 16 divides 16, 64, and 80. Because the leading coefficient is negative, pull out −16 so the leading term inside becomes positive.
Divide each term by −16. −16t² ÷ (−16) = t², 64t ÷ (−16) = −4t, and 80 ÷ (−16) = −5.
Write the factored form. −16t² + 64t + 80 = −16(t² − 4t − 5).
Factor the trinomial inside. Section 6 shows how; here t² − 4t − 5 = (t − 5)(t + 1). So −16t² + 64t + 80 = −16(t − 5)(t + 1).
Check by expanding. (t − 5)(t + 1) = t² + t − 5t − 5 = t² − 4t − 5, and −16 times that is −16t² + 64t + 80.

In words: the height formula now shows its building blocks. Next week, those blocks tell you the moment the ball lands.

Misconception check. Students often skip the GCF and reach straight for a pattern. With 3x² − 75 that leaves you stuck, because 3x² and 75 are not perfect squares. It is tempting because patterns feel faster. The cure: always ask "what do all the terms share?" first. Pulling out 3 gives 3(x² − 25), and Section 8 finishes it. An honesty note: a factoring with the GCF left inside, such as (3x − 15)(x + 5), multiplies out correctly but is not fully factored.

6. Factor trinomials that start with x squared

Now run FOIL backward. When you multiplied (x + 3)(x + 4), the last terms 3 and 4 multiplied to make 12, and the outer and inner products 4x and 3x added to make 7x. So to factor x² + 7x + 12, you hunt for two numbers that multiply to 12 and add to 7.

Here is the rule for a trinomial x² + bx + c, where the leading coefficient is 1: find two numbers whose product is c and whose sum is b. Those two numbers go into the two factors.

Worked example — positive terms. Factor x² + 7x + 12.

List the factor pairs of 12. 1 and 12, 2 and 6, 3 and 4.
Check the sums. 1 + 12 = 13, 2 + 6 = 8, 3 + 4 = 7. Only 3 and 4 add to 7.
Write the factors. x² + 7x + 12 = (x + 3)(x + 4).
Check by expanding. x² + 4x + 3x + 12 = x² + 7x + 12.

In words: this is the garden bed's area, taken apart into its two side lengths.

Worked example — a negative constant. Factor x² − 5x − 14.

Read the signs. The product is −14, which is negative, so one number is positive and one is negative. The sum is −5, so the bigger one is negative.
Try pairs. (−7)(2) = −14 and −7 + 2 = −5. That works.
Write the factors. x² − 5x − 14 = (x − 7)(x + 2).
Check by expanding. x² + 2x − 7x − 14 = x² − 5x − 14.

In words: the two numbers are −7 and 2.

Worked example — the shared polynomial, factored back. Factor x² − 6x + 5.

Read the signs. The product is 5, which is positive, so both numbers have the same sign. The sum is −6, so both are negative.
Try the pair. (−1)(−5) = 5 and −1 + (−5) = −6.
Write the factors. x² − 6x + 5 = (x − 1)(x − 5).

In words: you are back to the factors you multiplied in Section 3. Multiplying and factoring are the same road, traveled in opposite directions.

Misconception check. Students often find two numbers with the right product but the wrong signs, such as (x + 7)(x − 2) for x² − 5x − 14. It is tempting because 7 and 2 are the right numbers. But +7 − 2 = 5, not −5, so that factoring expands to x² + 5x − 14. The cure: check the sum with its signs, then expand your answer before you move on.

7. Factor with a leading coefficient: the ac method

When the x² term has a number in front, the hunt changes a little. Take 2x² + 7x + 3. You cannot look for a pair that multiplies to 3, because the 2 out front also takes part in every product. The ac method folds that 2 in.

The ac method is a way to factor ax² + bx + c when a is not 1: find two numbers that multiply to a times c and add to b, use them to split the middle term, and factor by grouping.

Worked example — the ac method. Factor 2x² + 7x + 3.

Name a, b, and c. a = 2, b = 7, and c = 3.
Find ac. ac = 2 · 3 = 6.
Find two numbers with product 6 and sum 7. 1 and 6 work, since 1 · 6 = 6 and 1 + 6 = 7.
Split the middle term. 7x = x + 6x, so 2x² + 7x + 3 = 2x² + x + 6x + 3.
Group the first two terms and the last two terms. x(2x + 1) + 3(2x + 1).
Pull out the shared binomial. (2x + 1)(x + 3).
Check by expanding. 2x² + 6x + x + 3 = 2x² + 7x + 3.

In words: 2x² + 7x + 3 = (2x + 1)(x + 3). Step 5 is grouping, which Section 9 treats on its own.

Misconception check. Students often look for a pair that multiplies to c = 3 and adds to 7, as if the leading 2 were not there. No such pair exists, so they decide the trinomial is prime. It is tempting because that hunt worked in Section 6. The cure: when a is not 1, multiply a and c first. The 2 has to be part of the hunt.

8. Spot a difference of squares, and leave a sum of squares alone

Section 4 showed that the patio's (x + 4)(x − 4) collapses to x² − 16, because the middle terms cancel. Run that backward and you get a fast factoring rule. Whenever you see one perfect square minus another, it factors into a sum times a difference.

A difference of squares is a binomial of the form a² − b², which always factors as (a + b)(a − b); for example, 9x² − 25 = (3x + 5)(3x − 5).
A sum of squares is a binomial of the form a² + b², such as x² + 16; with no common factor it cannot be factored over the real numbers.
A prime polynomial is a polynomial that cannot be written as a product of two polynomials of lower degree with real coefficients, other than by pulling out a constant; x² + 16 is an example.

Worked example — a difference of squares. Factor 9x² − 25.

Write each term as a square. 9x² = (3x)², and 25 = 5².
Use the pattern. (3x)² − 5² = (3x + 5)(3x − 5).
Check by expanding. 9x² − 15x + 15x − 25 = 9x² − 25.

In words: 9x² − 25 factors as (3x + 5)(3x − 5).

Worked example — GCF first, then the pattern. Factor 3x² − 75.

Pull out the GCF. 3 divides both terms, so 3x² − 75 = 3(x² − 25).
Spot the difference of squares inside. x² − 25 = x² − 5² = (x − 5)(x + 5).
Put it together. 3x² − 75 = 3(x − 5)(x + 5).

In words: the GCF step turned a stuck problem into a pattern you know.

Worked example — a sum of squares. Try to factor x² + 16.

Try two plus signs. (x + 4)(x + 4) = x² + 8x + 16. The middle term 8x should not be there.
Try a plus and a minus. (x + 4)(x − 4) = x² − 16. The sign of the 16 is wrong.
Conclude. Neither one gives you x² + 16, and there is no other way to arrange two binomials with real numbers.

In words: x² + 16 is a prime polynomial over the real numbers. Here is a second way to see it. The value of x² + 16 is always at least 16, because x² is never negative. A product (x − r)(x − s) with real r and s would equal 0 at x = r, and x² + 16 never does.

Misconception check. Students often factor x² + 16 as (x + 4)(x + 4) or as (x + 4)(x − 4). It is tempting because x² − 16 factors so neatly, and the plus sign is easy to overlook. The cure: expand your answer. One gives x² + 8x + 16, and the other gives x² − 16. Neither is x² + 16. A caveat for later courses: a sum of squares does factor if you allow numbers that are not real, but this course stays with real numbers.

9. Factor four terms by grouping, then check by expanding

Sometimes a polynomial has four terms and no single pattern fits. Grouping splits the job in two. You factor the first pair, factor the second pair, and hope the two leftovers match.

Factoring by grouping is a way to factor four terms: pair them, pull the greatest common factor out of each pair, and then pull out the binomial the two pairs share.

Worked example — grouping. Factor x³ + 2x² + 3x + 6.

Pair the terms. (x³ + 2x²) + (3x + 6).
Factor each pair. x³ + 2x² = x²(x + 2), and 3x + 6 = 3(x + 2).
Spot the shared binomial. Both pairs contain (x + 2).
Pull it out. x²(x + 2) + 3(x + 2) = (x + 2)(x² + 3).
Check by expanding. (x + 2)(x² + 3) = x³ + 3x + 2x² + 6 = x³ + 2x² + 3x + 6.

In words: x³ + 2x² + 3x + 6 = (x + 2)(x² + 3). The factor x² + 3 is a sum of squares in disguise, so it does not factor further over the real numbers.

Here is the order to try things in, every time you factor.

The order of checks for factoring a polynomial, with one example of each
Step	What to look for	Example
1	A greatest common factor	6x³ − 9x² = 3x²(2x − 3)
2	Two terms: a difference of squares	9x² − 25 = (3x + 5)(3x − 5)
3	Three terms, leading coefficient 1: product c, sum b	x² + 7x + 12 = (x + 3)(x + 4)
4	Three terms, leading coefficient not 1: the ac method	2x² + 7x + 3 = (2x + 1)(x + 3)
5	Four terms: grouping	x³ + 2x² + 3x + 6 = (x + 2)(x² + 3)
6	Expand to check	multiply the factors back out

In practice (your tools). Desmos will check any factoring you do. Type your original polynomial on line 1, such as y = x^3 + 2x^2 + 3x + 6, and your factored answer on line 2, such as y = (x + 2)(x^2 + 3). If you see only one curve, the two lines are the same polynomial. If two curves appear, expand by hand and find the slip.

Misconception check. Students often stop after step 2 of grouping and write x²(x + 2) + 3(x + 2) as the answer. It is tempting because each pair is factored. But that expression is still a sum of two pieces, not a product. The cure: a factored answer is one product. Pull out the shared binomial, and then expand to make sure you get the original back.

A reflection to choose from. Pick one option, (a) or (b), and write two or three sentences.

(a) Think of a rectangle in your own life that could grow in two directions, such as a room, a garden, a phone screen, or a sports field. Describe it with two binomial side lengths, and say what each of the four FOIL pieces stands for in the picture.
(b) Think about which factoring step you are most likely to skip: the GCF, the sign check, or the final expansion. Say why it is easy to skip, and describe one habit that would help you catch the slip.

Key terms

Polynomial — An expression built by adding terms, where each term is a number times a variable raised to a whole-number exponent, such as 3x² − 2x + 5. (first used: 1. Name the parts of a polynomial)
Term — One piece of a sum, separated from the next piece by a plus or minus sign; in 3x² − 2x + 5 the terms are 3x², −2x, and 5. (first used: 1. Name the parts of a polynomial)
Degree — The largest exponent on the variable in a polynomial; the degree of 3x² − 2x + 5 is 2. (first used: 1. Name the parts of a polynomial)
Leading coefficient — The number that multiplies the highest-power term of a polynomial; in 3x² − 2x + 5 the leading coefficient is 3. (first used: 1. Name the parts of a polynomial)
Constant term — The term of a polynomial with no variable in it, the plain number at the end when the terms are written from highest power to lowest. (first used: 1. Name the parts of a polynomial)
Monomial — A polynomial with exactly one term, such as 3x² or 7. (first used: 1. Name the parts of a polynomial)
Binomial — A polynomial with exactly two terms, such as x + 3 or 6x³ − 9x². (first used: 1. Name the parts of a polynomial)
Trinomial — A polynomial with exactly three terms, such as x² + 7x + 12. (first used: 1. Name the parts of a polynomial)
FOIL — A memory aid for multiplying two binomials: multiply the First terms, the Outer terms, the Inner terms, and the Last terms, then add the four products. (first used: 3. Multiply with FOIL and check at a value)
Special products — Three multiplication patterns worth knowing by sight: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², and (a + b)(a − b) = a² − b². (first used: 4. Use the three special products)
Perfect-square trinomial — A trinomial that is the square of a binomial, such as x² + 12x + 36 = (x + 6)²; its middle term is twice the product of the binomial's two terms. (first used: 4. Use the three special products)
Factoring — Rewriting a polynomial as a product of simpler polynomials; it undoes multiplication, and you check it by multiplying the factors back out. (first used: 5. Factor out the greatest common factor first)
Greatest common factor — The largest expression that divides every term of a polynomial, often written GCF; for 6x³ − 9x² it is 3x², so 6x³ − 9x² = 3x²(2x − 3). (first used: 5. Factor out the greatest common factor first)
Ac method — A way to factor ax² + bx + c when a is not 1: find two numbers that multiply to a times c and add to b, use them to split the middle term, and factor by grouping. (first used: 7. Factor with a leading coefficient: the ac method)
Difference of squares — A binomial of the form a² − b², which always factors as (a + b)(a − b); for example, 9x² − 25 = (3x + 5)(3x − 5). (first used: 8. Spot a difference of squares, and leave a sum of squares alone)
Sum of squares — A binomial of the form a² + b², such as x² + 16; with no common factor it cannot be factored over the real numbers. (first used: 8. Spot a difference of squares, and leave a sum of squares alone)
Prime polynomial — A polynomial that cannot be written as a product of two polynomials of lower degree with real coefficients, other than by pulling out a constant; x² + 16 is an example. (first used: 8. Spot a difference of squares, and leave a sum of squares alone)
Factoring by grouping — A way to factor four terms: pair them, pull the greatest common factor out of each pair, and then pull out the binomial the two pairs share. (first used: 9. Factor four terms by grouping, then check by expanding)

Summary

A polynomial is a sum of terms; its degree is the biggest exponent, and its leading coefficient rides on that term.
To subtract, flip the sign of every term in the second polynomial, then combine like terms.
Two binomials make four products; FOIL keeps track of them.
Check any product by evaluating both forms at one number.
The square of a binomial has a middle term, 2ab; never drop it.
Factor out the GCF first, every time.
For x² + bx + c, find two numbers with product c and sum b.
For ax² + bx + c, find two numbers with product ac and sum b, then group.
A difference of squares factors; a sum of squares is prime over the real numbers.
Expand every factored answer to check it.

Check your understanding

Work these before looking at the answers — they are ungraded, and the answers (with reasoning) are at the bottom of the page.

A school store tracks two price formulas. Subtract (4x² + 3x − 7) − (x² − 5x + 2) and check your answer at x = 1.
A square tablecloth is x inches on a side, and a 7-inch strip is cut from two neighboring edges, leaving a square x − 7 inches on a side. Expand (x − 7)² and check it at x = 10.
Factor x² + 2x − 24 and 5x² − 20 completely.
Factor 3x² + 10x + 8 by the ac method, and check by expanding.

Answers (worked reasoning — read after attempting)

Flip every sign in the second polynomial: 4x² + 3x − 7 − x² + 5x − 2. Combine: 4x² − x² = 3x², 3x + 5x = 8x, and −7 − 2 = −9. The difference is 3x² + 8x − 9. Check at x = 1: the original is (4 + 3 − 7) − (1 − 5 + 2) = 0 − (−2) = 2, and 3 + 8 − 9 = 2. If you flipped only the first sign, you would get 3x² − 2x − 5, which gives −4 at x = 1 and fails the check.
Use (a − b)² = a² − 2ab + b² with a = x and b = 7: x² − 2(x)(7) + 7² = x² − 14x + 49. Check at x = 10: (10 − 7)² = 3² = 9, and 100 − 140 + 49 = 9. Writing x² − 49 drops the middle term, and that gives 51 at x = 10, which fails.
For x² + 2x − 24, you need product −24 and sum 2. The pair 6 and −4 works, since 6 · (−4) = −24 and 6 + (−4) = 2. So x² + 2x − 24 = (x + 6)(x − 4); expanding gives x² − 4x + 6x − 24 = x² + 2x − 24. For 5x² − 20, pull out the GCF first: 5(x² − 4). Then x² − 4 is a difference of squares, so 5x² − 20 = 5(x + 2)(x − 2).
Here a = 3, b = 10, and c = 8, so ac = 24. You need product 24 and sum 10: 4 and 6 work. Split: 3x² + 4x + 6x + 8. Group: x(3x + 4) + 2(3x + 4) = (3x + 4)(x + 2). Check: 3x² + 6x + 4x + 8 = 3x² + 10x + 8. Hunting for a pair with product 8 and sum 10 finds nothing, which is the Section 7 trap.

Next up in this module: the Readings & Resources page, the lecture tutorial, the practice set, the quiz, and the discussion — and next week the zero-product property turns these factored forms into a method for solving quadratic equations.

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