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.