Wize Grade 11 Mathematics Textbook > Polynomials Factoring and Graphs

Factoring Special Trinomials

Perfect Squares
A polynomial is a perfect square if it is the result of a binomial that has been squared such as

(a+b)2=a2+2ab+b2(a−b)2=a2−2ab+b2\boxed{\begin{aligned}
(a+b)^2 &= a^2 + 2ab + b^2 \\
(a - b)^2 &= a^2 - 2ab + b^2
\end{aligned}}(a+b)2(a−b)2​=a2+2ab+b2=a2−2ab+b2​​
If you can recognize a polynomial in either of these forms, you can quickly use the formula to factor them. 

Example 1

Identify if the polynomial is a perfect square.  
If it is, write it in factored form.

1.  x2+4x+4x^2 + 4x + 4x2+4x+4

This is a perfect square where a=xa = xa=x and b=2b = 2b=2.
Factored form =(x+2)2=(x+2)^2=(x+2)2

2.  9x2−48x+649x^2 - 48x + 649x2−48x+64

This is a perfect square where a=3xa = 3xa=3x and b=−8b = -8b=−8.
Factored form =(3x−8)2=(3x-8)^2=(3x−8)2

3.  9x2+16x−49x^2 + 16x - 49x2+16x−4

This is not a perfect square.  
It can be factored, but not using the perfect squares formula. 

Difference of Squares
A polynomial is the difference of squares if it can be written as two terms squared and subtracted such as

(a+b)(a−b)=a2−b2\boxed{(a+b)(a-b) = a^2 - b^2}(a+b)(a−b)=a2−b2​
If you can recognize a polynomial in this form, you can quickly use the formula to factor it.

Example

Identify if the polynomial is the difference of squares.
If it is, write it in factored form.

1.  x2−4x^2 - 4x2−4

This is the difference of squares where a=xa =xa=x and b=2b = 2b=2.
Factored form =(x+2)(x−2)= (x+2)(x-2)=(x+2)(x−2)

2.  25x2−925x^2 - 925x2−9

This is the difference of squares where a=5xa = 5xa=5x and b=3b = 3b=3.
Factored form =(5x+3)(5x−3)= (5x + 3)(5x - 3)=(5x+3)(5x−3)

3.  9x2+169x^2 + 169x2+16

This is not the difference of squares.

Sum and Difference of Cubes
A polynomial is the sum or difference of cubes if it can be written as two terms cubed either added or subtracted such as

(a+b)(a2−ab+b2)=a3+b3(a−b)(a2+ab+b2)=a3−b3\boxed{
\begin{aligned}(a+b)(a^2 - ab + b^2) &= a^3 + b^3 \\
(a-b)(a^2 + ab + b^2) &= a^3 - b^3
\end{aligned}}(a+b)(a2−ab+b2)(a−b)(a2+ab+b2)​=a3+b3=a3−b3​​
If you can recognize polynomials in either of these forms, you can quickly use the formulas to factor them.

Example

Identify if the polynomial as the sum or difference of cubes.
If it is, write it in factored form.

1.  27x3+827x^3 +827x3+8

This is the sum of cubes where a=3xa =3xa=3x and b=2b = 2b=2.
Factored form =(3x+2)(9x2−6x+4)= (3x+2)(9x^2-6x + 4)=(3x+2)(9x2−6x+4)

2.  8x6−18x^6 - 18x6−1

This is the difference of cubes where a=2x2a = 2x^2a=2x2 and b=1b = 1b=1.
Factored form =(2x2−1)(4x4+2x2+1)= (2x^2 - 1)(4x^4 + 2x^2 + 1)=(2x2−1)(4x4+2x2+1)

3.  x3−9x^3 - 9x3−9

This is not the the sum or difference of cubes.

Wize Tip
To memorize this formula keep track of the pattern of a and b as well as how the sign matches up.  

(a  sign  b)=(a  same  b)(a2  opposite  ab  positive  b2)(a \text{ \ \colorFive{sign} \ } b) = ( a \text{ \ \colorFour{same} \ } b)(a^2 \text{ \ \colorTwo{opposite} \ } ab \text{ \ \colorOne{positive} \ }  b^2)(a  sign  b)=(a  same  b)(a2  opposite  ab  positive  b2)

Example:  Factoring Special Trinomials
For each of the polynomials below say if it is a special trinomial or not.  
If it is a special trinomial, use its formula to quickly factor it.  

1.  81x2−1681x^2 - 1681x2−16

This is the difference of squares where a=9xa = 9xa=9x and b=4b = 4b=4, using the formula we have

81x2−16=(9x+4)(9x−4)81x^2 - 16 = (9x + 4)(9x - 4)81x2−16=(9x+4)(9x−4)

2.  4x2+12x+94x^2 + 12x   + 94x2+12x+9

This is a perfect square where a=2xa = 2xa=2x and b=3b = 3b=3, using the formula we have

4x2+12x+9=(2x+3)24x^2 + 12x + 9 = (2x + 3)^24x2+12x+9=(2x+3)2

3.  25x2−10x+425x^2 - 10x +425x2−10x+4

This is not a special trinomial.  This polynomial can not be factored and is considered prime.  

4.  8x6+1258x^6 + 1258x6+125

This is the sum of cubes where a=2x2a = 2x^2a=2x2 and b=5b = 5b=5, using the formula we have

8x6+125=(2x+5)(4x2−10x+25)8x^6 + 125 = (2x + 5)(4x^2 - 10x + 25)8x6+125=(2x+5)(4x2−10x+25)

Practice: Factoring Special Trinomials
Identify the value of a and b that need to be cubed for this polynomial to be the sum of cubes.  
f(x)=27+8x3f(x) = 27 + 8x^3f(x)=27+8x3

a =

b =

I don't know

Practice: Factoring Special Trinomials
Match each special polynomial with its name. 

A.

9x2+42x+499x^2 + 42x + 499x2+42x+49

B.

25x2−425x^2 - 425x2−4

C.

8x3−18x^3 - 18x3−1

D.

x3+27x^3 + 27x3+27

Perfect square

Difference of squares

Sum of cubes

Difference of cubes

I don't know

Practice: Factoring Special Trinomials
All of the following polynomials are special trinomials.
Fill in the missing numbers to make the formulas correct.  

1.  Sum of cubes

(?)3−x3=(5−x)(25+5x+x2)( ? )^3 - x^3 = (5 - x)(25 + 5x + x^2)(?)3−x3=(5−x)(25+5x+x2)

2.  Perfect square

36x2−(?)x+25=(6x−5)236x^2 - ( ? )x + 25 = (6x - 5)^236x2−(?)x+25=(6x−5)2

3.  Difference of squares

49x2−(?)=(7x+6)(7x−6)49x^2 - (?) = (7x + 6)(7x - 6)49x2−(?)=(7x+6)(7x−6)

1. ? =

2. ? =

3. ? =

I don't know