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Wize High School Grade 11 Math Textbook > Factoring Polynomials

Common Factoring and Factoring by Grouping

Multiplying Any PolynomialsPrevious Section
Factoring Simple Trinomials y=x2+bx+cy=x^2+bx+cNext Section
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Common Factoring

When given a polynomial, we can use common factoring to "take out" (divide out) any common terms.

How to common factor?

  1. Identify the greatest common factor between all of the terms → GCF\colorbox{yellow}{GCF}
  2. Divide the entire polynomial by this greatest common factor → left over polynomial after dividing the GCF\colorbox{aqua}{left over polynomial after dividing the GCF}
  3. Rewrite the polynomial in factored form → GCF(left over polynomial after dividing the GCF)\colorbox{yellow}{GCF}(\colorbox{aqua}{left over polynomial after dividing the GCF})
Wize Tip
This is the "reverse" of multiplying a monomial by a polynomial!

We can always check our answer by expanding the factored form to see if we end up with the original polynomial


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Example 1
Identify the greatest common factor between the following terms, then factor each polynomial.

a) 3x−63x-6

1. Greatest common factor: 3\colorbox{yellow}{$3$}.

2. Dividing each term by 33: 3x3+−63=x−2\dfrac{3x}{\bct 3}+\dfrac{-6}{\bct 3}=\colorbox{aqua}{$x-2$}

3. The factored form: 3x−6=3(x−2)3x-6=\colorbox{yellow}{$3$}(\colorbox{aqua}{$x-2$})


Check:      3(x−2)=3(x)+3(−2)=3x−6  ✓\begin{aligned} &~~~~~3(x-2)\\ &=3(x)+3(-2)\\ &=3x-6~~\checkmark \end{aligned}



b) −3x−6-3x-6

Greatest common factor: −3\colorbox{yellow}{$-3$}.

Dividing each term by −3-3: −3x−3+−6−3=x+2\dfrac{-3x}{\bct {-3}}+\dfrac{-6}{\bct{-3}}=\colorbox{aqua}{$x+2$}

The factored form: −3x−6=−3(x+2)-3x-6=\colorbox{yellow}{$-3$}(\colorbox{aqua}{$x+2$})

Check:    −3(x+2)=−3(x)−3(2)=−3x−6  ✓\begin{aligned} &~~~-3(x+2)\\ &=-3(x)-3(2)\\ &=-3x-6~~\checkmark \end{aligned}



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c) 3x2−63x^2-6

Greatest common factor: 3\colorbox{yellow}{$3$}.

Dividing each term by 33: 3x23+−63=x2−2\dfrac{3x^2}{\bct {3}}+\dfrac{-6}{\bct {3}}=\colorbox{aqua}{$x^2-2$}

The factored form: 3x2−6=3(x2−2)3x^2-6=\colorbox{yellow}{$3$}(\colorbox{aqua}{$x^2-2$})

Check:      3(x2−2)=3(x2)+3(−2)=3x2−6  ✓\begin{aligned} &~~~~~3(x^2-2)\\ &=3(x^2)+3(-2)\\ &=3x^2-6~~\checkmark \end{aligned}



d) −3x2−6-3x^2-6

Greatest common factor: −3\colorbox{yellow}{$-3$}.

Dividing each term by −3-3: −3x2−3+−6−3=x2+2\dfrac{-3x^2}{\bct {-3}}+\dfrac{-6}{\bct {-3}}=\colorbox{aqua}{$x^2+2$}

The factored form: −3x2−6=−3(x2+2)-3x^2-6=\colorbox{yellow}{$-3$}(\colorbox{aqua}{$x^2+2$})

Check:     −3(x2+2)=−3(x2)−3(2)=−3x2−6  ✓\begin{aligned} &~~~~-3(x^2+2)\\ &=-3(x^2)-3(2)\\ &=-3x^2-6~~\checkmark \end{aligned}

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Example 2
Factor the following polynomials

a) 3x2−6x3x^2-6x

Greatest common factor: 3x\colorbox{yellow}{$3x$}.

Dividing each term by 3x3x: 3x23x+−6x3x=x−2\dfrac{3x^2}{\bct {3x}}+\dfrac{-6x}{\bct {3x}}=\colorbox{aqua}{$x-2$}

The factored form: 3x2−6x=3x(x−2)3x^2-6x=\colorbox{yellow}{$3x$}(\colorbox{aqua}{$x-2$})

Check:      3x(x−2)=3x(x)+3x(−2)=3x2−6x  ✓\begin{aligned} &~~~~~3x(x-2)\\ &=3x(x)+3x(-2)\\ &=3x^2-6x~~\checkmark \end{aligned}



b) −3x2−6x-3x^2-6x

Greatest common factor: −3x\colorbox{yellow}{$-3x$}.

Dividing each term by −3x-3x: −3x2−3x+−6x−3x=x+2\dfrac{-3x^2}{\bct {-3x}}+\dfrac{-6x}{\bct {-3x}}=\colorbox{aqua}{$x+2$}

The factored form: −3x2−6x=−3x(x+2)-3x^2-6x=\colorbox{yellow}{$-3x$}(\colorbox{aqua}{$x+2$})

Check:     −3x(x+2)=−3x(x)−3x(+2)=−3x2−6x  ✓\begin{aligned} &~~~~-3x(x+2)\\ &=-3x(x)-3x(+2)\\ &=-3x^2-6x~~\checkmark \end{aligned}

Practice: Greatest Common Factors

Identify the greatest common factors between the following terms.

a) 14x14x and 77

b) 2525 and −25x-25x

c) 4x24x^2 and 88

d) 12x212x^2 and −4x-4x

e) −24x-24x and −9x2-9x^2
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Example: Common Factoring

Factor the following polynomials
a) 3x2−6x+93x^2-6x+9

Watch Out!
The GCF must be a factor of ALL of the terms!

Even though xx is a common factor of the first two terms 3x23x^2 and −6x-6x, it is NOT a common factor of the last term+9+9, so it is NOT a GCF of all of the terms!

1. Greatest common factor: 3\colorbox{yellow}{$3$}.

2. Dividing each term by 33: 3x23+−6x3+93=x2−2x+3\dfrac{3x^2}{\bct 3}+\dfrac{-6x}{\bct 3}+\dfrac{9}{\bct 3}=\colorbox{aqua}{$x^2-2x+3$}

3. The factored form: 3x2−6x+9=3(x2−2x+3)3x^2-6x+9=\colorbox{yellow}{$3$}(\colorbox{aqua}{$x^2-2x+3$})

Check:      3(x2−2x+3)=3(x2)+3(−2x)+3(3)=3x2−6x+9  ✓\begin{aligned} &~~~~~3(x^2-2x+3)\\ &=3(x^2)+3(-2x)+3(3)\\ &=3x^2-6x+9~~\checkmark \end{aligned}


b) −3x2−6x+9-3x^2-6x+9

1. Greatest common factor: −3\colorbox{yellow}{$-3$}.

2. Dividing each term by −3-3: −3x2−3+−6x−3+9−3=x2+2x−3\dfrac{-3x^2}{\bct {-3}}+\dfrac{-6x}{\bct {-3}}+\dfrac{9}{\bct {-3}}=\colorbox{aqua}{$x^2+2x-3$}

3. The factored form: −3x2−6x+9=−3(x2+2x−3)-3x^2-6x+9=\colorbox{yellow}{$-3$}(\colorbox{aqua}{$x^2+2x-3$})

Check:      −3(x2+2x−3)=−3(x2)−3(2x)−3(−3)=−3x2−6x+9  ✓\begin{aligned} &~~~~~-3(x^2+2x-3)\\ &=-3(x^2)-3(2x)-3(-3)\\ &=-3x^2-6x+9~~\checkmark \end{aligned}

*In later sections, you will learn how to further factor this polynomial!
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c) −3x2a+6xa2-3x^2a+6xa^2

1. Greatest common factor: −3xa\colorbox{yellow}{$-3xa$}.

2. Dividing each term by −3xa-3xa: −3x2a−3xa+6xa2−3xa=x−2a\dfrac{-3x^2a}{\bct {-3xa}}+\dfrac{6xa^2}{\bct {-3xa}}=\colorbox{aqua}{$x-2a$}

3. The factored form: −3x2a+6xa2=−3xa(x−2a)-3x^2a+6xa^2=\colorbox{yellow}{$-3xa$}(\colorbox{aqua}{$x-2a$})

Check:      −3xa(x−2a)=−3xa(x)−3xa(−2a)=−3x2a+6xa2  ✓\begin{aligned} &~~~~~-3xa(x-2a)\\ &=-3xa(x)-3xa(-2a)\\ &=-3x^2a+6xa^2~~\checkmark \end{aligned}


d) −2x2a2+8xa2b-2x^2a^2+8xa^2b

1. Greatest common factor: −2xa2\colorbox{yellow}{$-2xa^2$}.

2. Dividing each term by −2xa2-2xa^2: −2x2a2−2xa2+8xa2b−2xa2=x−4b\dfrac{-2x^2a^2}{\bct {-2xa^2}}+\dfrac{8xa^2b}{\bct {-2xa^2}}=\colorbox{aqua}{$x-4b$}

3. The factored form: −2x2a2+8xa2b=−2xa2(x−4b)-2x^2a^2+8xa^2b=\colorbox{yellow}{$-2xa^2$}(\colorbox{aqua}{$x-4b$})

Check:      −2xa2(x−4b)=−2xa2(x)−2xa2(−4b)=−2x2a2+8xa2b  ✓\begin{aligned} &~~~~~-2xa^2(x-4b)\\ &=-2xa^2(x)-2xa^2(-4b)\\ &=-2x^2a^2+8xa^2b~~\checkmark \end{aligned}


e) 4a2b2c2−8abc2+10a2b2c4a^2b^2c^2-8abc^2+10a^2b^2c

1. Greatest common factor: 2abc\colorbox{yellow}{$2abc$}.

2. Dividing each term by 2abc2abc: 4a2b2c22abc+−8abc22abc+10a2b2c2abc=2abc−4c+5ab\dfrac{4a^2b^2c^2}{\bct {2abc}}+\dfrac{-8abc^2}{\bct {2abc}}+\dfrac{10a^2b^2c}{\bct{2abc}}=\colorbox{aqua}{$2abc-4c+5ab$}

3. The factored form: 4a2b2c2−8abc2+10a2b2c=2abc(2abc−4c+5ab)4a^2b^2c^2-8abc^2+10a^2b^2c=\colorbox{yellow}{$2abc$}(\colorbox{aqua}{$2abc-4c+5ab$})

Check:      2abc(2abc−4c+5ab)=2abc(2abc)+2abc(−4c)+2abc(5ab)=4a2b2c2−8abc2+10a2b2c  ✓\begin{aligned} &~~~~~2abc(2abc-4c+5ab)\\ &=2abc(2abc)+2abc(-4c)+2abc(5ab)\\ &=4a^2b^2c^2-8abc^2+10a^2b^2c~~\checkmark \end{aligned}

Practice: Common Factoring

Factor the following polynomials.

a) −4x2y−10x2y2-4x^2y-10x^2y^2

b) −5x2ya+10x2y2a2−5xya-5x^2ya+10x^2y^2a^2-5xya

c) 6x2y−3xy+9z6x^2y-3xy+9z

Practice: Common Factoring

Identify the greatest common factor between terms, then factor the polynomial.
3x(x−1)+5(x−1)3x(x-1)+5(x-1)

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Example: Grouping then Common Factoring

Wize Tip
Sometimes, you won't be able to find a common factor among all terms in a polynomial. Try grouping the terms to see if there's a common factor among the smaller group of terms.

Factor the following polynomials.

a) 3x2+3x−4xy−4y3x^2+3x-4xy-4y

Grouping the first two terms and the last two terms together:
[3x2+3x]⏟Group 1+[−4xy−4y]⏟Group 2\underbrace{[3x^2+3x]}_\text{Group 1}+\underbrace{[-4xy-4y]}_\text{Group 2}

Group 1:
1. Greatest common factor: 3x\colorbox{yellow}{$3x$}.

2. Dividing each term by 3x3x: 3x23x+3x3x=x+1\dfrac{3x^2}{\bct {3x}}+\dfrac{3x}{\bct {3x}}=\colorbox{aqua}{$x+1$}

3. The factored form: 3x2+3x=3x(x+1)3x^2+3x=\colorbox{yellow}{$3x$}(\colorbox{aqua}{$x+1$})

Group 2:
1. Greatest common factor: −4y\colorbox{yellow}{$-4y$}.

2. Dividing each term by −4y-4y: −4xy−4y+−4y−4y=x+1\dfrac{-4xy}{\bct {-4y}}+\dfrac{-4y}{\bct {-4y}}=\colorbox{aqua}{$x+1$}

3. The factored form: −4xy−4y=−4y(x+1)-4xy-4y=\colorbox{yellow}{$-4y$}(\colorbox{aqua}{$x+1$})


So, factoring in pairs, we get 3x(x+1)−4y(x+1)3x(x+1)-4y(x+1).

Notice that there's another common factor of (x+1)(x+1) between these two terms! Factoring further, we see that 3x2+3x−4xy−4y=(x+1)(3x−4y)3x^2+3x-4xy-4y=\boxed{(x+1)(3x-4y)}



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b) mx−nx+my−nymx-nx+my-ny

Grouping the first two terms and the last two terms together:
[mx−nx]⏟Group 1+[my−ny]⏟Group 2\underbrace{[mx-nx]}_\text{Group 1}+\underbrace{[my-ny]}_\text{Group 2}

Group 1:
1. Greatest common factor: x\colorbox{yellow}{$x$}.

2. Dividing each term by xx: mxx+−nxx=m−n\dfrac{mx}{\bct {x}}+\dfrac{-nx}{\bct {x}}=\colorbox{aqua}{$m-n$}

3. The factored form: mx−nx=x(m−n)mx-nx=\colorbox{yellow}{$x$}(\colorbox{aqua}{$m-n$})

Group 2:
1. Greatest common factor: y\colorbox{yellow}{$y$}.

2. Dividing each term by yy: myy+−nyy=m−n\dfrac{my}{\bct {y}}+\dfrac{-ny}{\bct {y}}=\colorbox{aqua}{$m-n$}

3. The factored form: my−ny=y(m−n)my-ny=\colorbox{yellow}{$y$}(\colorbox{aqua}{$m-n$})


So, factoring in pairs, we get x(m−n)+y(m−n)x(m-n)+y(m-n).

Notice that there's another common factor of (m−n)(m-n) between these two terms! Factoring further, we see that mx−nx+my−ny=(m−n)(x+y)mx-nx+my-ny=\boxed{(m-n)(x+y)},

Practice: Common Factoring by Grouping

Factor the polynomial 2x2−3+6x−x2x^2-3+6x-x.

Practice: Common Factoring

Factor the following polynomials.

a) 8x2−12x+288x^2-12x+28

b) −5x2−10xy+15y-5x^2-10xy+15y

c) 6xd2−2x2d2−4xd6xd^2-2x^2d^2-4xd

d) −6a2b2c+15ab2c2−27abc2-6a^2b^2c+15ab^2c^2-27abc^2

e) 4x(x−1)−2(x−1)4x(x-1)-2(x-1)

f) 5x(3x+y)−xy(3x+y)−6y(3x+y)5x(3x+y)-xy(3x+y)-6y(3x+y)