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Introduction to Factoring Polynomials

Find the greatest common factor of two or more expressions

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

In The Language of Algebra we factored numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.

First we will find the greatest common factor of two numbers.

Example

Try it.

Find the greatest common factor of \(24\) and \(36.\)

Solution
Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form.Factor 24 and 36.
Step 2: List all factors--matching common factors in a column.
In each column, circle the common factors.Circle the 2, 2, and 3 that are shared by both numbers.
Step 3: Bring down the common factors that all expressions share.Bring down the 2, 2, 3 and then multiply.
Step 4: Multiply the factors.The GCF of 24 and 36 is 12.

Notice that since the GCF is a factor of both numbers, \(24\) and \(36\) can be written as multiples of \(12.\)

\[\begin{array}{l}24=12\cdot 2 \\ 36=12\cdot 3\end{array}\]

In the previous example, we found the greatest common factor of constants. The greatest common factor of an algebraic expression can contain variables raised to powers along with coefficients. We summarize the steps we use to find the greatest common factor.

Example

Try it.

Find the greatest common factor of \(5x\ \text{and}\ 15.\)

Solution
Factor each number into primes.
Circle the common factors in each column.
Bring down the common factors.
The GCF of 5x and 15 is 5.

In the examples so far, the greatest common factor was a constant. In the next two examples we will get variables in the greatest common factor.

Condensed: the full section is in OpenStax Prealgebra 2e.

Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, \(12\) as \(2\cdot 6\ \text{or}\ 3\cdot 4\text{),}\) in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:

\[\begin{array}{lll}2(x & + & 7)\ \text{factors} \\ 2\cdot x & + & 2\cdot 7 \\ 2x & + & 14\ \text{product}\end{array}\]

Here, we will start with a product, like \(2x+14,\) and end with its factors, \(2(x+7).\) To do this we apply the Distributive Property “in reverse”.

The form on the left is used to multiply. The form on the right is used to factor.

So how do we use the Distributive Property to factor a polynomial? We find the GCF of all the terms and write the polynomial as a product!

Example

Try it.

Factor: \(2x+14.\)

Solution
Step 1: Find the GCF of all the terms of the polynomial.Find the GCF of 2x and 14.
Step 2: Rewrite each term as a product using the GCF.Rewrite 2x and 14 as products of their GCF, 2.
\(2x=2⋅x\)
\(14=2⋅7\)
Step 3: Use the Distributive Property 'in reverse' to factor the expression.\(2(x+7)\)
Step 4: Check by multiplying the factors.Check:

Notice that in , we used the word factor as both a noun and a verb:

\[\begin{array}{llll}\text{Noun} & & & 7\ \text{is a factor of}\ 14 \\ \text{Verb} & & & \text{factor}\ 2\ \text{from}\ 2x+14\end{array}\]
Example

Try it.

Factor: \(3a+3.\)

Solution
Rewrite each term as a product using the GCF.
Use the Distributive Property 'in reverse' to factor the GCF.
Check by multiplying the factors to get the original polynomial.

The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.

Example

Try it.

Factor: \(12x-60.\)

Solution
Rewrite each term as a product using the GCF.
Factor the GCF.
Check by multiplying the factors.

Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.

Condensed: the full section is in OpenStax Prealgebra 2e.

Key Concepts

  • Find the greatest common factor.
    1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
    2. List all factors, matching common factors in a column. In each column, circle the common factors.
    3. Bring down the common factors that all expressions share.
    4. Multiply the factors.
  • Distributive Property
    • If \(a\), \(b\), \(c\) are real numbers, then
      \(a(b+c)=ab+ac\) and \(ab+ac=a(b+c)\)
  • Factor the greatest common factor from a polynomial.
    1. Find the GCF of all the terms of the polynomial.
    2. Rewrite each term as a product using the GCF.
    3. Use the Distributive Property ‘in reverse’ to factor the expression.
    4. Check by multiplying the factors.

Chapter Practice Test

In the following exercises, simplify each expression.

In the following exercises, factor the greatest common factor from each polynomial.

In the following exercises, simplify, and write your answer in decimal form.

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Praxis (40)

Versuchen Sie zuerst jeden auf Papier. Zeigen Sie die Antwort zu überprüfen; überprüfte können im Löser für jeden Schritt geöffnet werden.

  1. Factor \(56\) into primes.

    Die Antwort aufzeigen

    \(2⋅2⋅2⋅7\)

  2. Multiply: \(-3(6a+11).\)

    Die Antwort aufzeigen

    \(-18a-33\)

  3. Multiply: \(4{x}^{2}({x}^{2}+3x-1).\)

    Die Antwort aufzeigen

    \(4{x}^{4}+12{x}^{3}-4{x}^{2}\)

  4. Find the greatest common factor of \(24\) and \(36.\)

    Die Antwort aufzeigen
    Step 1: Factor each coefficient into primes. Write all variables with exponents in expanded form.Factor 24 and 36.
    Step 2: List all factors--matching common factors in a column.
    In each column, circle the common factors.Circle the 2, 2, and 3 that are shared by both numbers.
    Step 3: Bring down the common factors that all expressions share.Bring down the 2, 2, 3 and then multiply.
    Step 4: Multiply the factors.The GCF of 24 and 36 is 12.

    Notice that since the GCF is a factor of both numbers, \(24\) and \(36\) can be written as multiples of \(12.\)

    \[\begin{array}{l}24=12\cdot 2 \\ 36=12\cdot 3\end{array}\]

  5. Find the greatest common factor: \(54,36.\)

    Die Antwort aufzeigen

    18

  6. Find the greatest common factor: \(48,80.\)

    Die Antwort aufzeigen

    16

  7. Find the greatest common factor of \(5x\ \text{and}\ 15.\)

    Die Antwort aufzeigen
    Factor each number into primes.
    Circle the common factors in each column.
    Bring down the common factors.
    The GCF of 5x and 15 is 5.

  8. Find the greatest common factor: \(7y,\ 14.\)

    Die Antwort aufzeigen

    7

  9. Find the greatest common factor: \(22,\ 11m.\)

    Die Antwort aufzeigen

    11

  10. Find the greatest common factor of \(12{x}^{2}\) and \(18{x}^{3}.\)

    Die Antwort aufzeigen
    Factor each coefficient into primes and write
    the variables with exponents in expanded form.
    Circle the common factors in each column.
    Bring down the common factors.
    Multiply the factors.
    \(\text{The GCF of}\ 12{x}^{2}\ \text{and}\ 18{x}^{3}\ \text{is}\ 6{x}^{2}\)

  11. Find the greatest common factor: \(16{x}^{2},\ 24{x}^{3}.\)

    Die Antwort aufzeigen

    8x2

  12. Find the greatest common factor: \(27{y}^{3},\ 18{y}^{4}.\)

    Die Antwort aufzeigen

    9y3

  13. Find the greatest common factor of \(14{x}^{3},\ 8{x}^{2},\ 10x.\)

    Die Antwort aufzeigen
    Factor each coefficient into primes and write
    the variables with exponents in expanded form.
    Circle the common factors in each column.
    Bring down the common factors.
    Multiply the factors.
    \(\text{The GCF of}\ 14{x}^{3}\ \text{and}\ 8{x}^{2}\text{, and}10x\ \text{is}\ 2x\)

  14. Find the greatest common factor: \(21{x}^{3},\ 9{x}^{2},\ 15x.\)

    Die Antwort aufzeigen

    3x

  15. Find the greatest common factor: \(25{m}^{4},\ 35{m}^{3},\ 20{m}^{2}.\)

    Die Antwort aufzeigen

    5m2

  16. Factor: \(2x+14.\)

    Die Antwort aufzeigen
    Step 1: Find the GCF of all the terms of the polynomial.Find the GCF of 2x and 14.
    Step 2: Rewrite each term as a product using the GCF.Rewrite 2x and 14 as products of their GCF, 2.
    \(2x=2⋅x\)
    \(14=2⋅7\)
    Step 3: Use the Distributive Property 'in reverse' to factor the expression.\(2(x+7)\)
    Step 4: Check by multiplying the factors.Check:

  17. Factor: \(4x+12.\)

    Die Antwort aufzeigen

    4(x + 3)

  18. Factor: \(6a+24.\)

    Die Antwort aufzeigen

    6(a + 4)

  19. Factor: \(3a+3.\)

    Die Antwort aufzeigen
    Rewrite each term as a product using the GCF.
    Use the Distributive Property 'in reverse' to factor the GCF.
    Check by multiplying the factors to get the original polynomial.

  20. Factor: \(9a+9.\)

    Die Antwort aufzeigen

    9(a + 1)

  21. Factor: \(11x+11.\)

    Die Antwort aufzeigen

    11(x + 1)

  22. Factor: \(12x-60.\)

    Die Antwort aufzeigen
    Rewrite each term as a product using the GCF.
    Factor the GCF.
    Check by multiplying the factors.

  23. Factor: \(11x-44.\)

    Die Antwort aufzeigen

    11(x − 4)

  24. Factor: \(13y-52.\)

    Die Antwort aufzeigen

    13(y − 4)

  25. Factor: \(3{y}^{2}+6y+9.\)

    Die Antwort aufzeigen
    Rewrite each term as a product using the GCF.
    Factor the GCF.
    Check by multiplying.

  26. Factor: \(4{y}^{2}+8y+12.\)

    Die Antwort aufzeigen

    4(y2 + 2y + 3)

  27. Factor: \(6{x}^{2}+42x-12.\)

    Die Antwort aufzeigen

    6(x2 + 7x − 2)

  28. Factor: \(6{x}^{2}+5x.\)

    Die Antwort aufzeigen
    \(6{x}^{2}+5x\)
    Find the GCF of \(6{x}^{2}\) and \(5x\) and the math that goes with it.
    Rewrite each term as a product.
    Factor the GCF.\(x(6x+5)\)
    Check by multiplying.
    \(x(6x+5)\)
    \(x⋅6x+x⋅5\)
    \(6{x}^{2}+5x✓\)

  29. Factor: \(9{x}^{2}+7x.\)

    Die Antwort aufzeigen

    x(9x + 7)

  30. Factor: \(5{a}^{2}-12a.\)

    Die Antwort aufzeigen

    a(5a − 12)

  31. Factor: \(4{x}^{3}-20{x}^{2}.\)

    Die Antwort aufzeigen
    Rewrite each term.
    Factor the GCF.
    Check.

  32. Factor: \(2{x}^{3}+12{x}^{2}.\)

    Die Antwort aufzeigen

    2x2(x + 6)

  33. Factor: \(6{y}^{3}-15{y}^{2}.\)

    Die Antwort aufzeigen

    3y2(2y − 5)

  34. Factor: \(21{y}^{2}+35y.\)

    Die Antwort aufzeigen
    Find the GCF of \(21{y}^{2}\) and \(35y\)
    Rewrite each term.
    Factor the GCF.

  35. Factor: \(18{y}^{2}+63y.\)

    Die Antwort aufzeigen

    9y(2y + 7)

  36. Factor: \(32{k}^{2}+56k.\)

    Die Antwort aufzeigen

    8k(4k + 7)

  37. Factor: \(14{x}^{3}+8{x}^{2}-10x.\)

    Die Antwort aufzeigen

    Previously, we found the GCF of \(14{x}^{3},\ 8{x}^{2},\ \text{and}\ 10x\) to be \(2x.\)

    \(14{x}^{3}+8{x}^{2}-10x\)
    Rewrite each term using the GCF, 2x.
    Factor the GCF.\(2x(7{x}^{2}+4x-5)\)

  38. Factor: \(18{y}^{3}-6{y}^{2}-24y.\)

    Die Antwort aufzeigen

    6y(3y2 − y − 4)

  39. Factor: \(16{x}^{3}+8{x}^{2}-12x.\)

    Die Antwort aufzeigen

    4x(4x2 + 2x − 3)

  40. Factor: \(-9y-27.\)

    Die Antwort aufzeigen
    When the leading coefficient is negative, the GCF will be negative. Ignoring the signs of the terms, we first find the GCF of 9y and 27 is 9.
    Since the expression −9y−27 has a negative leading coefficient, we use −9 as the GCF.
    \(-9y-27\)
    Rewrite each term using the GCF.
    Factor the GCF.\(-9(y+3)\)
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Wie man: Introduction to Factoring Polynomials

  1. Find the greatest common factor of two or more expressions
  2. Factor the greatest common factor from a polynomial
  3. Factor each coefficient into primes. Write all variables with exponents in expanded form.
  4. List all factors—matching common factors in a column. In each column, circle the common factors.
  5. Bring down the common factors that all expressions share.
  6. Multiply the factors.
  7. Find the GCF of all the terms of the polynomial.
  8. Rewrite each term as a product using the GCF.

Fragen, die die Leute stellen

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

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