Wize High School Grade 10 Math Textbook > Factoring Polynomials

The Idea behind Common Factoring
Let's say you have the following 3 pencil cases:


It's clear that there are items in common (the same) between all 3 pencil cases. How can we simplify this picture using numbers?

Wize Tip
When thinking about common factoring, you want to "take out" (divide out) the largest common item as possible!

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Factoring with Numbers
Given any number nnn, its factors are divisors of nnn (meaning numbers that "go into nnn very nicely").

*Note: We usually only consider integer factors!

Example 1
Given the number 24, its factors are -1, 1, -2, 2, -3, 3, -4, 4, -6, 6, -8, 8, -12, 12, -24, and 24.

So, we can rewrite 24 as follows
24=24=24=     -1 (-24)     
24=24=24=     1(24)     
24=24=24=     -2 (-12)     
24=24=24=     2 (12)     
24=24=24=     -3 (-8)     
24=24=24=     3 (8)     
...

Example 2
Identify the greatest common factor between the following terms, then rewrite the expression by dividing out the greatest common factor.

a) 5+25=5+25=5+25=          5 (1 + 5)           


b) 6−15=6-15=6−15=          3 (2 - 5)           or          -3 (-2 + 5)          


c) 8−20=8-20=8−20=          4 (-2 - 5)           or          -4 (-2 + 5)               


d) −8−20=-8-20=−8−20=          -4 (2 + 5)           

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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?
Identify the greatest common factor between all of the terms → GCF\colorbox{yellow}{GCF}GCF​
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}left over polynomial after dividing the GCF​
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})GCF​(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-63x−6

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

2. Dividing each term by 333: 3x3+−63=x−2\dfrac{3x}{\bct 3}+\dfrac{-6}{\bct 3}=\colorbox{aqua}{$x-2$}33x​+3−6​=x−2​

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

b) −3x−6-3x-6−3x−6

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

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

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

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c) 3x2−63x^2-63x2−6

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

Dividing each term by 333: 3x23+−63=x2−2\dfrac{3x^2}{\bct {3}}+\dfrac{-6}{\bct {3}}=\colorbox{aqua}{$x^2-2$}33x2​+3−6​=x2−2​

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

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}​     3(x2−2)=3(x2)+3(−2)=3x2−6  ✓​

d) −3x2−6-3x^2-6−3x2−6

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

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

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

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Example 2
Factor the following polynomials

a) 3x2−6x3x^2-6x3x2−6x

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

Dividing each term by 3x3x3x: 3x23x+−6x3x=x−2\dfrac{3x^2}{\bct {3x}}+\dfrac{-6x}{\bct {3x}}=\colorbox{aqua}{$x-2$}3x3x2​+3x−6x​=x−2​

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

b) −3x2−6x-3x^2-6x−3x2−6x

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

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

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

Practice: Greatest Common Factors
Identify the greatest common factors between the following terms.

a) 14x14x14x and 777

b) 252525 and −25x-25x−25x

c) 4x24x^24x2 and 888

d) 12x212x^212x2 and −4x-4x−4x

e) −24x-24x−24x and −9x2-9x^2−9x2

I don't know

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

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

Even though xxx is a common factor of the first two terms 3x23x^23x2 and −6x-6x−6x, it is NOT a common factor of the last term+9+9+9, so it is NOT a GCF of all of the terms!

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

2. Dividing each term by 333: 3x23+−6x3+93=x2−2x+3\dfrac{3x^2}{\bct 3}+\dfrac{-6x}{\bct 3}+\dfrac{9}{\bct 3}=\colorbox{aqua}{$x^2-2x+3$}33x2​+3−6x​+39​=x2−2x+3​

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

b) −3x2−6x+9-3x^2-6x+9−3x2−6x+9

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

2. Dividing each term by −3-3−3: −3x2−3+−6x−3+9−3=x2+2x−3\dfrac{-3x^2}{\bct {-3}}+\dfrac{-6x}{\bct {-3}}+\dfrac{9}{\bct {-3}}=\colorbox{aqua}{$x^2+2x-3$}−3−3x2​+−3−6x​+−39​=x2+2x−3​

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

*In later sections, you will learn how to further factor this polynomial!

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c) −3x2a+6xa2-3x^2a+6xa^2−3x2a+6xa2

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

2. Dividing each term by −3xa-3xa−3xa: −3x2a−3xa+6xa2−3xa=x−2a\dfrac{-3x^2a}{\bct {-3xa}}+\dfrac{6xa^2}{\bct {-3xa}}=\colorbox{aqua}{$x-2a$}−3xa−3x2a​+−3xa6xa2​=x−2a​

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

d) −2x2a2+8xa2b-2x^2a^2+8xa^2b−2x2a2+8xa2b

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

2. Dividing each term by −2xa2-2xa^2−2xa2: −2x2a2−2xa2+8xa2b−2xa2=x−4b\dfrac{-2x^2a^2}{\bct {-2xa^2}}+\dfrac{8xa^2b}{\bct {-2xa^2}}=\colorbox{aqua}{$x-4b$}−2xa2−2x2a2​+−2xa28xa2b​=x−4b​

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

e) 4a2b2c2−8abc2+10a2b2c4a^2b^2c^2-8abc^2+10a^2b^2c4a2b2c2−8abc2+10a2b2c

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

2. Dividing each term by 2abc2abc2abc: 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}{$2abc-4c+5ab$}2abc4a2b2c2​+2abc−8abc2​+2abc10a2b2c​=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}{$2abc-4c+5ab$})4a2b2c2−8abc2+10a2b2c=2abc​(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}​     2abc(2abc−4c+5ab)=2abc(2abc)+2abc(−4c)+2abc(5ab)=4a2b2c2−8abc2+10a2b2c  ✓​

Practice: Common Factoring
Factor the following polynomials.

a) −4x2y−10x2y2-4x^2y-10x^2y^2−4x2y−10x2y2

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

c) 6x2y−3xy+9z6x^2y-3xy+9z6x2y−3xy+9z

I don't know

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)3x(x−1)+5(x−1)

The greatest common factor is

The factored form is

I don't know

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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-4y3x2+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[3x2+3x]​​+Group 2[−4xy−4y]​​

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

2. Dividing each term by 3x3x3x: 3x23x+3x3x=x+1\dfrac{3x^2}{\bct {3x}}+\dfrac{3x}{\bct {3x}}=\colorbox{aqua}{$x+1$}3x3x2​+3x3x​=x+1​

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

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

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

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

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

Notice that there's another common factor of (x+1)(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)}3x2+3x−4xy−4y=(x+1)(3x−4y)​

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b) mx−nx+my−nymx-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[mx−nx]​​+Group 2[my−ny]​​

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

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

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

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

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

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

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

Notice that there's another common factor of (m−n)(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)}mx−nx+my−ny=(m−n)(x+y)​,

Practice: Common Factoring by Grouping
Factor the polynomial 2x2−3+6x−x2x^2-3+6x-x2x2−3+6x−x.

I don't know

Practice: Common Factoring
Factor the following polynomials.

a) 8x2−12x+288x^2-12x+288x2−12x+28

b) −5x2−10xy+15y-5x^2-10xy+15y−5x2−10xy+15y

c) 6xd2−2x2d2−4xd6xd^2-2x^2d^2-4xd6xd2−2x2d2−4xd

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

e) 4x(x−1)−2(x−1)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)5x(3x+y)−xy(3x+y)−6y(3x+y)

I don't know