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Factoring Polynomials

Factor expressions using the GCF, trinomial methods, the difference of squares, and grouping.

Practice this skill 25 min Factor polynomials Leer en español
Key idea

What you'll learn

Factoring is multiplication run in reverse. When you multiply, you start with pieces and build a bigger expression: (x + 2)(x + 3) = x^2 + 5x + 6. When you factor, you start with the big expression and find the pieces that were multiplied to make it.

In this lesson you will learn the four factoring tools the GED expects:

  1. Pulling out the greatest common factor (GCF).
  2. Factoring trinomials like x^2 + 7x + 12 and 6x^2 + 7x - 20.
  3. Spotting the difference of squares, a^2 - b^2.
  4. Factoring by grouping when there are four terms.

Why it matters on the GED. Factoring shows up in several ways. Some questions simply ask, "Which expression is equivalent to x^2 - 5x - 14?" Others hide it inside a bigger job: simplifying a fraction with x in it, solving a quadratic equation, or finding the missing side of a rectangle when you know its area. If you can factor quickly and check your answer, many algebra questions become short.

Real life. Factoring is how you "undo" an area. If a patio has an area of x^2 + 9x + 20 square feet and one side is x + 4 feet, factoring tells you the other side is x + 5 feet.

Words to know

  • Factor (noun) — a number or expression that is multiplied by another. In 3 \cdot 5 = 15, both 3 and 5 are factors of 15.
  • Factor (verb) — to rewrite an expression as a product. Factoring 6x + 9 gives 3(2x + 3).
  • Term — one piece of an expression separated by + or - signs. x^2 - 4x + 3 has three terms.
  • Coefficient — the number in front of a variable. In -4x, the coefficient is -4.
  • Constant — a term with no variable. In x^2 - 4x + 3, the constant is 3.
  • Greatest common factor (GCF) — the largest number and highest power of each variable that divides every term. The GCF of 12x^3 and 18x^2 is 6x^2.
  • Binomial — an expression with two terms, such as x - 7.
  • Trinomial — an expression with three terms, such as x^2 + 8x + 15.
  • Leading coefficient — the coefficient of the highest power. In 6x^2 - x - 15 it is 6.
  • Difference of squares — one perfect square minus another, such as x^2 - 49 or 4x^2 - 9.
  • Perfect square trinomial — a trinomial that equals a binomial squared, such as x^2 + 6x + 9 = (x + 3)^2.
  • Factor by grouping — splitting four terms into two pairs, factoring each pair, then pulling out the shared binomial.
  • Prime (cannot be factored) — an expression with no factors except 1 and itself, such as x^2 + 4 using whole numbers.
  • Factored completely — no factor can be broken down any further.

All Math words in the glossary

Tool 1: Pull out the greatest common factor

Always check for a GCF first. It makes every other step easier.

How to find the GCF

  1. Find the largest number that divides every coefficient.
  2. For each variable that appears in every term, take the smallest exponent you see.
  3. Write the GCF outside parentheses. Inside, write what is left after dividing each term by the GCF.
ExpressionGCFFactored
6x + 933(2x + 3)
10x^2 - 15x5x5x(2x - 3)
4x^3 - 8x^2 + 12x4x4x(x^2 - 2x + 3)
3x^2 + 3x3x3x(x + 1)
Area model for factoring out a GCF: a rectangle of height 3 is split into a 3 by 2x part with area 6x and a 3 by 3 part with area 9. The height 3 is the GCF and the total width 2x + 3 is what is left, so 6x + 9 = 3(2x + 3).6x92x332x + 36x + 9 = 3(2x + 3)

Look at the last row. When a term is exactly the GCF (3x \div 3x), it leaves a 1, not a zero. Writing 3x(x) would lose a term.

Negative leading term. If the first term is negative, it is common to pull out a negative GCF so the inside starts with a positive term. Every sign inside flips:

-2x^2 + 6x = -2x(x - 3)

Check: -2x \cdot x = -2x^2 and -2x \cdot (-3) = +6x ✓.

A binomial can be the common factor. In x(x + 2) + 5(x + 2), both terms contain the same "chunk" (x + 2). Pull it out like any other factor:

x(x + 2) + 5(x + 2) = (x + 2)(x + 5)

This idea is the heart of factoring by grouping (Tool 4).

Check every factoring by multiplying back. Distribute the GCF. If you get the original expression, you are right.

Tool 2: Factor trinomials

Case 1: leading coefficient is 1, x^2 + bx + c.

Find two numbers that multiply to c and add to b. Call them m and n. Then

x^2 + bx + c = (x + m)(x + n)

Example: x^2 + 7x + 12. Pairs that multiply to 12: 1 \cdot 12, 2 \cdot 6, 3 \cdot 4. Only 3 + 4 = 7. So x^2 + 7x + 12 = (x + 3)(x + 4).

Let the signs guide you.

Sign of cSign of bThe two numbers are…Example
++both positivex^2 + 8x + 15 = (x + 3)(x + 5)
+-both negativex^2 - 8x + 15 = (x - 3)(x - 5)
-+opposite signs; bigger one positivex^2 + 2x - 15 = (x + 5)(x - 3)
--opposite signs; bigger one negativex^2 - 2x - 15 = (x - 5)(x + 3)

The area model below shows why this works. The four boxes are x^2, 3x, 2x and 6. The two middle boxes add to 5x, and the corner box is 2 \cdot 3 = 6.

Case 2: leading coefficient is not 1, ax^2 + bx + c (the "ac method").

  1. Multiply a \cdot c.
  2. Find two numbers that multiply to ac and add to b.
  3. Split the middle term into those two pieces.
  4. Factor by grouping.

Example: 2x^2 + 7x + 3. Here ac = 2 \cdot 3 = 6. The numbers 6 and 1 multiply to 6 and add to 7.

2x^2 + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)

Check: (2x + 1)(x + 3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3 ✓.

Area model for 2x squared + 7x + 3. Columns are x and 3; rows are 2x and 1. The four boxes are 2x squared, 6x, x and 3. The middle boxes 6x and x add to 7x, and 6 times 1 equals a times c = 6. So 2x squared + 7x + 3 = (2x + 1)(x + 3).2x²6xx3x32x1x + 32x + 1a · c = 2 · 3 = 66x + x = 7x(6 · 1 = 6, 6 + 1 = 7)2x² + 7x + 3 = (2x + 1)(x + 3)

Shortcut for multiple-choice questions. You do not always have to factor from scratch. Multiply out each answer choice (FOIL: First, Outer, Inner, Last) and see which one matches. Checking only the first and last terms is not enough: for 6x^2 + 7x - 20, the choices (6x - 4)(x + 5) and (3x - 4)(2x + 5) both start with 6x^2 and end with -20, but only (3x - 4)(2x + 5) has the right middle term, +7x. The other gives +26x.

Area model: a rectangle split into four parts. Top row: x by x square labeled x squared and x by 3 rectangle labeled 3x. Bottom row: 2 by x rectangle labeled 2x and 2 by 3 rectangle labeled 6. Side lengths are x + 3 across the top and x + 2 down the side, so x squared + 5x + 6 = (x + 2)(x + 3).x²3x2x6x3x2x² + 5x + 6 = (x + 2)(x + 3)

Area model for x^2 + 5x + 6. The side lengths of the big rectangle, x + 2 and x + 3, are the factors.

Tool 3: Special patterns

Difference of squares. When you see one perfect square minus another, use

a^2 - b^2 = (a - b)(a + b)
Difference of squares picture. Left: a square of side a with a smaller b by b square removed from one corner, leaving an L-shape of area a squared minus b squared. The L is cut into a blue piece and a green piece. Right: the green piece is moved to the end of the blue piece, making a rectangle a + b long and a − b tall. So a squared − b squared = (a − b)(a + b), for example x squared − 49 = (x − 7)(x + 7).b²removedaabba² − b²movea + ba − b(a − b)(a + b)a² − b² = (a − b)(a + b) e.g. x² − 49 = (x − 7)(x + 7)
  • x^2 - 49 = (x - 7)(x + 7)
  • 9x^2 - 25 = (3x - 5)(3x + 5), because 9x^2 = (3x)^2 and 25 = 5^2
  • x^2 - \tfrac{1}{4} = (x - \tfrac{1}{2})(x + \tfrac{1}{2})

Perfect squares worth knowing on sight: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and any even power such as x^2, x^4, x^6.

A sum of squares does not factor. x^2 + 9 is prime with whole numbers. (x + 3)^2 = x^2 + 6x + 9, which has a middle term, so it is not equal to x^2 + 9.

Perfect square trinomials.

a^2 + 2ab + b^2 = (a + b)^2 \qquad a^2 - 2ab + b^2 = (a - b)^2

Test: are the first and last terms perfect squares, and is the middle term twice their square roots multiplied? For x^2 - 10x + 25: \sqrt{x^2} = x, \sqrt{25} = 5, and 2 \cdot x \cdot 5 = 10x ✓. So x^2 - 10x + 25 = (x - 5)^2.

Perfect square trinomial picture: a square with side a + b is split into an a by a square (a squared), two a by b rectangles (ab each) and a b by b square (b squared). Its area is a squared + 2ab + b squared, which equals (a + b) squared. Example: (x + 3) squared = x squared + 6x + 9.a²ababb²ababSide: a + bArea: a² + 2ab + b²Example: a = x, b = 3(x + 3)² = x² + 6x + 9(a + b)² = a² + 2ab + b²

Tool 4: Grouping, and factoring completely

Factor by grouping (four terms).

  1. Group the first two terms and the last two terms.
  2. Pull the GCF out of each group.
  3. If the two parentheses match, pull that binomial out.

Example: x^3 + 3x^2 + 2x + 6

= x^2(x + 3) + 2(x + 3) = (x + 3)(x^2 + 2)

Watch the sign in the second group. For x^3 - 5x^2 - 4x + 20, pull -4 (not +4) out of the second pair so the parentheses match:

x^2(x - 5) - 4(x - 5) = (x - 5)(x^2 - 4) = (x - 5)(x - 2)(x + 2)

Factoring completely. A GED question that says "factor completely" wants every piece broken down. Use this checklist in order:

  1. Pull out the GCF.
  2. Count the terms. Two terms: difference of squares? Three terms: trinomial? Four terms: grouping?
  3. Look at each new factor. Can it be factored again?

Example: 2x^2 - 18 = 2(x^2 - 9) = 2(x - 3)(x + 3).

Example: x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4). The factor x^2 + 4 is a sum of squares, so it stays.

Connection to solving. Later you will use factoring to solve equations like x^2 - 5x - 14 = 0: factor to (x - 7)(x + 2) = 0, then each factor can equal zero, so x = 7 or x = -2.

Formula

Patterns to memorize (not on the formula sheet)

a^2 - b^2 = (a - b)(a + b) a^2 + 2ab + b^2 = (a + b)^2 a^2 - 2ab + b^2 = (a - b)^2 x^2 + bx + c = (x + m)(x + n) \quad \text{where } m \cdot n = c \text{ and } m + n = b
  • a and b stand for any terms (numbers, variables, or both), such as a = 3x and b = 5.
  • m and n are the two "magic numbers" that multiply to the constant and add to the middle coefficient.

None of these factoring patterns is on the GED formula sheet. The sheet gives the quadratic formula, but not the factoring rules. You must memorize them. The good news: you can always check a factoring by multiplying it back out.

Worked example

Example 1 (easy): Pull out the GCF

Factor completely: 12x^3 + 18x^2.

Box model for Example 1: a rectangle with height 6x squared (the GCF) split into two parts. The part of width 2x has area 12x cubed and the part of width 3 has area 18x squared, so 12x cubed + 18x squared = 6x squared times (2x + 3).12x³18x²2x36x²2x + 3GCF12x³ + 18x² = 6x²(2x + 3)
  1. Numbers: the largest number dividing 12 and 18 is 6.
  2. Variables: both terms have x; the smallest power is x^2.
  3. GCF = 6x^2.
  4. Divide each term: 12x^3 \div 6x^2 = 2x and 18x^2 \div 6x^2 = 3.

Answer: 6x^2(2x + 3)

Check: 6x^2 \cdot 2x + 6x^2 \cdot 3 = 12x^3 + 18x^2 ✓

Worked example

Example 2: A trinomial with a negative constant

Factor: x^2 - 3x - 28.

Example 2. Left: a box model with columns x and −7 and rows x and +4; the boxes are x squared, −7x, 4x and −28. Right: a product–sum X with −28 on top (multiply to), −3 on the bottom (add to), and the two numbers −7 and 4 at the sides. −7x + 4x = −3x, so x squared − 3x − 28 = (x − 7)(x + 4).x²−7x4x−28x−7x+4−28−3−74multiply toadd to−7x + 4x = −3x so x² − 3x − 28 = (x − 7)(x + 4)
  1. We need two numbers that multiply to -28 and add to -3.
  2. The product is negative, so the signs are opposite. The sum is negative, so the bigger number is negative.
  3. Try pairs: -28 \cdot 1 (sum -27), -14 \cdot 2 (sum -12), -7 \cdot 4 (sum -3) ✓.

Answer: (x - 7)(x + 4)

Check: x^2 + 4x - 7x - 28 = x^2 - 3x - 28 ✓

Worked example

Example 3 (real world, find the missing side): A patio

A rectangular patio has an area of x^2 + 9x + 20 square feet. Its width is x + 4 feet. Write an expression for its length.

Patio drawn to scale with x = 10: a rectangle whose width is x + 4 feet and whose length is x + 5 feet, split into boxes x squared, 5x, 4x and 20. The boxes add to x squared + 9x + 20. With x = 10 the patio is 14 by 15 feet, area 210 square feet.x²5x4x20x5x4length = x + 5width = x + 4Area = x² + 9x + 20 = (x + 4)(x + 5)Check x = 10: 14 × 15 = 210 sq ft
  1. Area = length × width, so the area expression must factor with (x + 4) as one factor.
  2. Two numbers that multiply to 20 and add to 9: 4 and 5.
  3. x^2 + 9x + 20 = (x + 4)(x + 5).
  4. The width is x + 4, so the other factor is the length.

Answer: the length is x + 5 feet.

Check: Let x = 10. Area = 100 + 90 + 20 = 210. Width = 14, length = 15, and 14 \cdot 15 = 210 ✓

Worked example

Example 4 (GED-hard): Leading coefficient not 1

Factor: 6x^2 - x - 15.

Example 4. Left: a box model with columns 3x and −5 and rows 2x and +3; the boxes are 6x squared, −10x, 9x and −15. Right: a product–sum X: the two numbers −10 and 9 multiply to a times c = −90 and add to −1. So 6x squared − x − 15 = (2x + 3)(3x − 5).6x²−10x9x−153x−52x+3−90−1−109multiply toadd toa · c = 6 · (−15)−10x + 9x = −x so 6x² − x − 15 = (2x + 3)(3x − 5)
  1. a \cdot c = 6 \cdot (-15) = -90.
  2. Find two numbers that multiply to -90 and add to -1: -10 and 9 (because -10 \cdot 9 = -90 and -10 + 9 = -1).
  3. Split the middle term: 6x^2 - 10x + 9x - 15.
  4. Group: 2x(3x - 5) + 3(3x - 5).
  5. Pull out the shared binomial: (3x - 5)(2x + 3).

Answer: (2x + 3)(3x - 5)

Check: (2x + 3)(3x - 5) = 6x^2 - 10x + 9x - 15 = 6x^2 - x - 15 ✓

Worked example

Example 5 (real world, GED-hard): The dimensions of a crate

The volume of a storage crate is x^3 + 2x^2 - 9x - 18 cubic feet. Factor the expression completely to find expressions for the three dimensions.

Storage crate for Example 5, drawn to scale for x = 5: a box with length x + 3 (8 ft), height x − 3 (2 ft) and depth x + 2 (7 ft). Volume x cubed + 2x squared − 9x − 18 = (x + 2)(x − 3)(x + 3); at x = 5 this is 7 times 2 times 8 = 112 cubic feet.x + 3x − 3x + 2V = x³ + 2x² − 9x − 18 = (x + 2)(x − 3)(x + 3)Drawn for x = 5: 7 × 2 × 8 = 112 cubic feet
  1. Four terms, so try grouping: (x^3 + 2x^2) + (-9x - 18).
  2. Factor each group. Pull -9 from the second group so the parentheses match: x^2(x + 2) - 9(x + 2).
  3. Pull out (x + 2): (x + 2)(x^2 - 9).
  4. x^2 - 9 is a difference of squares: (x - 3)(x + 3).

Answer: (x + 2)(x - 3)(x + 3), so the dimensions are x + 2, x - 3 and x + 3 feet.

Check: Let x = 5. Original: 125 + 50 - 45 - 18 = 112. Factored: 7 \cdot 2 \cdot 8 = 112 ✓

Calculator tip

Calculator tip: check, don't factor

The TI-30XS MultiView cannot factor for you, and most factoring questions are faster by hand. Use the calculator only to check an answer by plugging in a number.

  1. Pick a test value such as x = 5. Avoid 0 and 1: they can make wrong answers look right.
  2. Evaluate the original expression. For 6x^2 - x - 15 at x = 5, type 6 × 5 x² − 5 − 15 enter → 130.
  3. Evaluate the answer choice. For (2x + 3)(3x - 5), type ( 2 × 5 + 3 ) ( 3 × 5 − 5 ) enter → 13 \cdot 10 = 130 ✓.
  4. For negative test values, use the (−) key, not the subtraction key, and wrap them in parentheses before squaring: ( (−) 2 ) x² gives 4.

When not to use it: If the numbers are small, mental FOIL is quicker than typing. Save the calculator for messy coefficients.

Common mistake

Common mistakes

  • Stopping before the GCF is complete. Wrong: 10x^2 - 15x = 5(2x^2 - 3x). The inside still has a common x. Right: 5x(2x - 3).
  • Matching only the product, or only the sum. For x^2 + 8x + 15, the pair 2, 6 adds to 8 but multiplies to 12; the pair 1, 15 multiplies to 15 but adds to 16. You need both: 3 and 5.
  • Sign mix-ups. (x - 3)(x - 5) = x^2 - 8x + 15, not x^2 + 8x + 15. If the middle term is positive and the constant is positive, both signs are +.
  • Treating a difference of squares like a perfect square. Wrong: x^2 - 9 = (x - 3)^2. Right: (x - 3)(x + 3). The square (x - 3)^2 has a middle term -6x.
  • Trying to factor a sum of squares. Wrong: x^2 + 4 = (x + 2)(x + 2). That product is x^2 + 4x + 4. A sum of squares does not factor.
  • Not factoring completely. 3x^4 - 48 = 3(x^2 - 4)(x^2 + 4) is not finished, because x^2 - 4 = (x - 2)(x + 2).
Test strategy

GED test strategy

  • Multiply the choices back out. On a "which is equivalent" question, the fastest path is often to FOIL each answer and compare all three terms.
  • Eliminate by signs. Look at the constant term. If it is negative, the two factors must have opposite signs, so any choice with two + or two - signs is out.
  • Plug in a number. Choose x = 5 (or any value other than 0 or 1). Evaluate the original and each choice. Only the correct factoring matches.
  • Read the question twice. "Factor completely" means every factor must be broken down. "Which is a factor of…" means you only need one piece.
  • Missing-side questions: if a rectangle's area and one side are given, the other side is the other factor.
  • Time: a basic trinomial should take under a minute. If you are stuck, switch to plugging in.

Check yourself

Factor polynomialsNo calculator

Which expression is the complete factorization of 10x^2 - 15x?

Answer choices
Hint

Find the largest number that divides 10 and 15, and the highest power of x that is in both terms.

Check yourself

Factor polynomialsNo calculator

Which expression is equivalent to x^2 + 8x + 15?

Answer choices
Hint

Look for two numbers that multiply to the constant term and add to the middle coefficient.

Summary

  • GCF first: largest number and smallest power of each shared variable; check by distributing.
  • x^2 + bx + c: find two numbers that multiply to c and add to b. Use the sign table.
  • ax^2 + bx + c: multiply a \cdot c, split the middle term, then group.
  • Difference of squares: a^2 - b^2 = (a - b)(a + b). A sum of squares does not factor.
  • Four terms: factor by grouping, watching the sign of the second group.
  • Always check by multiplying back or plugging in a number. These patterns are not on the formula sheet.
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