Factoring Completely

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Factoring completely means writing a polynomial as a product of factors that can’t be factored any more. Take out the GCF first. Then use the number of terms to pick a method, and keep going until every factor is prime. For example, 2x^2 - 18 = 2(x + 3)(x - 3).

Before this lesson: You should know factoring out the GCF, factoring trinomials, factoring by grouping, the difference of squares and perfect square trinomials.

What Does It Mean to Factor Completely?

Think about prime factorization. You can write 60 = 2 \cdot 30, but 30 still breaks down. The full answer is 60 = 2 \cdot 2 \cdot 3 \cdot 5.

Polynomials work the same way. 2x^2 - 18 = 2(x^2 - 9) is factored, but not completely. The part x^2 - 9 still factors. The complete answer is 2(x + 3)(x - 3).

A polynomial that can’t be factored with integers is called prime. Examples are x + 3, x^2 + 4 and x^2 + x + 1. You are done when every factor is prime.

The Factoring Strategy

A flowchart. Step 1: factor out the GCF. Step 2: count the terms. Three branches follow: 2 terms, try the difference of squares; 3 terms, try a perfect square trinomial or the trinomial methods; 4 terms, try grouping. Step 3: check each factor and factor again if you can.
GCF first. Then let the number of terms pick the method. Then check every factor again.
Number of termsWhat to tryExample
2Difference of squares, a^2 - b^2x^2 - 25 = (x + 5)(x - 5)
3Perfect square trinomial, or find two numbers (the AC method if a is not 1)x^2 + 7x + 12 = (x + 3)(x + 4)
4Factor by groupingx^3 + 2x^2 + 3x + 6 = (x + 2)(x^2 + 3)

Steps for factoring completely

  1. Factor out the GCF of all the terms. If the first term is negative, you can factor out a negative.
  2. Count the terms in what is left.
  3. Use the method that fits: 2 terms, 3 terms or 4 terms.
  4. Look at each factor. If one can be factored again, factor it.
  5. Check by multiplying all the factors.

Worked Examples

Example 1: GCF, then a difference of squares

Factor 3x^2 - 12 completely.

The GCF is 3. What is left, x^2 - 4, is a difference of squares.

\begin{aligned} 3x^2 - 12 &= 3(x^2 - 4) \\ &= 3(x + 2)(x - 2) \end{aligned}

Example 2: GCF, then a perfect square trinomial

Factor 2x^3 + 16x^2 + 32x completely.

The GCF is 2x. The trinomial left over has ends x^2 and 16 = 4^2, and 2 \cdot x \cdot 4 = 8x. So it is a perfect square.

\begin{aligned} 2x^3 + 16x^2 + 32x &= 2x(x^2 + 8x + 16) \\ &= 2x(x + 4)^2 \end{aligned}

Example 3: GCF, then a trinomial with a not 1

Factor 4x^3 - 10x^2 - 6x completely.

The GCF is 2x. That leaves 2x^2 - 5x - 3. Here ac = -6, and -6 and 1 multiply to -6 and add to -5.

\begin{aligned} 4x^3 - 10x^2 - 6x &= 2x(2x^2 - 5x - 3) \\ &= 2x(2x^2 - 6x + x - 3) \\ &= 2x[2x(x - 3) + 1(x - 3)] \\ &= 2x(2x + 1)(x - 3) \end{aligned}

Example 4: Grouping, then a difference of squares

Factor x^3 + 2x^2 - 9x - 18 completely.

There is no GCF, and there are 4 terms, so group them. The factor x^2 - 9 is a difference of squares, so keep going.

\begin{aligned} x^3 + 2x^2 - 9x - 18 &= x^2(x + 2) - 9(x + 2) \\ &= (x + 2)(x^2 - 9) \\ &= (x + 2)(x + 3)(x - 3) \end{aligned}

Example 5: Word problem

A box has a volume of 2x^3 + 10x^2 + 12x cubic inches. Factor completely to find expressions for its three dimensions. Then find them when x = 3.

\begin{aligned} 2x^3 + 10x^2 + 12x &= 2x(x^2 + 5x + 6) \\ &= 2x(x + 2)(x + 3) \end{aligned}

The dimensions are 2x, x + 2 and x + 3 inches. When x = 3, they are 6, 5 and 6 inches. The volume is 6 \cdot 5 \cdot 6 = 180 cubic inches. Check: 2(27) + 10(9) + 12(3) = 54 + 90 + 36 = 180.

Common Mistakes

MistakeExample of the errorFix
Stopping after the GCF3x^2 - 12 = 3(x^2 - 4)x^2 - 4 still factors: 3(x + 2)(x - 2).
Losing the GCF3x^2 - 12 = (x + 2)(x - 2)Keep the 3. (x + 2)(x - 2) = x^2 - 4, not 3x^2 - 12.
Skipping the GCF2x^2 + 10x + 12 = (2x + 4)(x + 3)2x + 4 still has a factor of 2. Take it out first: 2(x + 2)(x + 3).
Not factoring a second timex^4 - 16 = (x^2 + 4)(x^2 - 4)x^2 - 4 factors again: (x^2 + 4)(x + 2)(x - 2).
Factoring a sum of squaresx^2 + 4 = (x + 2)(x - 2)(x + 2)(x - 2) = x^2 - 4. The sum x^2 + 4 is prime.

How to Know You Are Done

Before you stop, run through this checklist.

QuestionIf yes
Do all the terms of any factor share a number or variable?Factor out that GCF.
Is any factor a difference of squares?Factor it as a sum times a difference.
Is any trinomial factor a perfect square or a factorable trinomial?Factor it.
Does any factor have 4 terms?Try grouping.

If every answer is no, the polynomial is factored completely. A factor like x^2 + 9 or x^2 + 2x + 5 can stay, because it is prime. Check the final answer by multiplying the factors back together.

Factoring completely is the key step for solving quadratic equations by factoring, which comes in Unit 9.

Practice Problems

  1. Factor 5x^2 - 45 completely.
  2. Factor 3x^2 + 18x + 27 completely.
  3. Factor x^3 - 25x completely.
  4. Factor 2x^2 + 14x + 24 completely.
  5. Factor 4x^2 - 4x - 24 completely.
  6. Factor 6x^3 + 3x^2 - 30x completely.
  7. Factor x^3 + 5x^2 - 4x - 20 completely.
  8. Factor 2x^4 - 32 completely.
  9. Factor 3x^2 + 12 completely.
  10. A box has a volume of 3x^3 + 21x^2 + 30x cubic feet. Factor completely to find its dimensions. Then find the dimensions and the volume when x = 2.
Show answers
  1. 5(x^2 - 9) = 5(x + 3)(x - 3)
  2. 3(x^2 + 6x + 9) = 3(x + 3)^2
  3. x(x^2 - 25) = x(x + 5)(x - 5)
  4. 2(x^2 + 7x + 12) = 2(x + 3)(x + 4)
  5. 4(x^2 - x - 6) = 4(x - 3)(x + 2)
  6. 3x(2x^2 + x - 10) = 3x(2x + 5)(x - 2)
  7. x^2(x + 5) - 4(x + 5) = (x + 5)(x^2 - 4) = (x + 5)(x + 2)(x - 2)
  8. 2(x^4 - 16) = 2(x^2 + 4)(x^2 - 4) = 2(x^2 + 4)(x + 2)(x - 2)
  9. 3(x^2 + 4). The sum x^2 + 4 is prime, so you are done.
  10. 3x(x^2 + 7x + 10) = 3x(x + 2)(x + 5). When x = 2: 6, 4 and 7 feet, and 6 \cdot 4 \cdot 7 = 168 cubic feet.

FAQ

What does it mean to factor completely?

It means to keep factoring until no factor can be broken down any more with integers. Every factor in the answer is either a number, a variable or a prime polynomial.

What should you always do first when factoring?

Look for a GCF and take it out. It makes the numbers smaller and often shows a pattern, like a difference of squares, that was hidden.

How do you know when a polynomial is factored completely?

Check each factor. If none has a common factor, none is a difference of squares, and no trinomial or four-term factor can be factored, you are done.

Is (2x + 4)(x + 3) factored completely?

No. The factor 2x + 4 has a GCF of 2. The complete answer is 2(x + 2)(x + 3).

Summary

  • Factoring completely means every factor is prime.
  • Always take out the GCF first.
  • Count the terms: 2 terms, try a difference of squares; 3 terms, try a perfect square or the trinomial methods; 4 terms, try grouping.
  • Check each factor and factor again if you can.
  • Keep the GCF in the answer, and check by multiplying.

Previous lesson: Perfect Square Trinomials

Next lesson: Simplifying Square Roots

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