Factoring of Special Polynomials - Questions With Detailed Solutions

How to use special polynomial forms to factor other polynomials? Grade 11 math questions are presented along with detailed Solutions and explanations. We will use five special polynomial forms.

1) Difference of two squares

a 2 - b 2 = (a - b)(a + b)

Examples: Factor the polynomial.

16 x 2 - 9 y 2

Solution:

Note that 16 x 2 = (4 x) 2 and 9 y 2 = (3 y) 2

We can write

16 x 2 - 9 y 2 = (4 x) 2 - (3 y) 2

Now that we have written the given polynomial as the the difference of two squares, we use formula above to factor the given polynomial as follows:

16 x 2 - 9 y 2 = (4 x) 2 - (3 y) 2 = (4 x - 3 y)(4 x + 3 y)


2) Trinomial Perfect Square

a)
a 2 + 2 a b + b 2 = (a + b) 2

b)
a 2 - 2 a b + b 2 = (a - b) 2

Examples: Factor the polynomials.

4 x 2 + 20 x y + 25 y 2

Solution:

Note that the monomials making the given polynomial may be written as follows:

4 x 2 = (2 x) 2, 20 x y = 2(2 x)(5 y) and 25 y 2 = (5 y) 2.

We now write the given polynomial as follows

4 x 2 + 10 x y + 25 y 2 = (2 x) 2 + 2(2 x)(5 y) + (5 y) 2

Use the formula
a 2 + 2 a b + b 2 = (a + b) 2 to write the given polynomial as a square as follows:

4 x 2 + 20 x y + 25 y 2 = (2 x) 2 + 2(2 x)(5 y) + (5 y) 2 = (2 x + 5 y) 2


Example: Factor the polynomials.

1 - 6 x + 9 x 2

Solution:

Note that the monomials making the given polynomial may be written as follows:

1 = 1 2, - 6 x = - 2(3)x and 9 x 2 = (3 x) 2.

The given polynomial may be written as follows

1 - 6 x + 9 x 2 = 1 2 - 2(3) x + (3 x) 2

Use the formula
a 2 - 2 a b + b 2 = (a - b) 2 to write the given polynomial as a square as follows:

1 - 6 x + 9 x 2 = 1 2 - 2(3) x + (3 x) 2 = (1 - 3 x) 2



3)
Difference of two cubes

a 3 - b 3 = (a - b)(a 2 + ab + b 2)

Example: Factor the polynomial.

8 - 27 x 3

Solution:

Note that the monomials making the given polynomial may be written as follows:

8 = (2) 3 and 27 x 3 = (3 x) 3

The given polynomial may now be written as follows

8 - 27 x 3 = (2) 3 - (3 x) 3

Use the formula
a 3 - b 3 = (a - b)(a 2 + ab + b 2) to write the given polynomial in factored as follows:

8 - 27 x 3 = (2) 3 - (3 x) 3 = (2 - 3 x)( (2) 2 + (2)(3x) + (3 x) 2) = (2 - 3 x)(9 x 2 + 6x + 4)


4)
Sum of two cubes

a 3 + b 3 = (a + b)(a 2 - ab + b 2)

Example: Factor the polynomial.

8 y 3 + 1

Solution:

The two monomials making the given polynomial may be written as follows:

8 y 3 = (2 y) 3 and 1 = (1) 3

The polynomial to factor may now be written as follows

8 y 3 + 1 = (2 y) 3 + (1) 3

Use the formula
a 3 + b 3 = (a + b)(a 2 - ab + b 2) to write the given polynomial in factored as follows:

8 y 3 + 1 = (2 y) 3 + (1) 3 = (2 y + 1)( (2 y) 2 - (2 y)(1) + (1) 2) = (2 y + 1)(4 y 2 - 2 y + 1)

Factor the following special polynomials

Detailed Solutions and explanations to these questions.

a) - 25 x 2 + 9

b) 16 y 4 - x 4

c) 36 y 2 - 60 x y + 25 x 2

d) (1/2) x 2 + x + (1/2)

e) - y 3 - 64

f) x 6 - 1

Detailed Solutions and explanations to these questions.

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Updated: 20 January 2017 (A Dendane)