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Special Polynomials

Special Polynomials

Certain polynomials can be factored by finding a pattern . This section deals with four special
patterns for factoring polynomials: difference of squares , difference of cubes, sum of cubes, and
perfect squares

Difference of Squares
The difference of squares pattern can be identified by looking at the polynomial. It must be a
binomial, the first term must be a variable to the second power (a.k.a. squared) and a constant
term must be subtracted from it. There is no first- order variable term in a difference-of-squares
polynomial. The formula a^2 - b^2 = (a - b)(a + b) is how a difference of squares polynomial is

Example: Factor x^2 – 25.
This binomial has its first term is x^2, a second-order monomial. The only other term is 25,
just a constant. This means x^2 – 25 can be factored using the difference of squares pattern, so
x^2 – 25 = (x)^2 – (5)^2 = (x – 5)(x + 5).
To check, we can multiply the factored form back together using the FOIL method :
(x – 5)(x + 5) = x^2 + 5x – 5x – 25 = x^2 – 25.

Example: Factor 9x^2 – 36.
This binomial’s highest order monomial is 9x^2; the other monomial is the constant 36, so we
can factor 9x^2 – 36 using the difference of squares method:

Note: There is no sum of squares factorization; that is, we can’t factor a^2 + b^2.

Difference of Cubes
To find the pattern for the difference of cubes, the polynomial to factor must be a binomial, the
first term must be a variable to the third power (a.k.a. cubed) and a constant term must be
subtracted from it. There cannot be any first- or second-order variable terms in a difference-of-cubes
polynomial. The formula a^3 - b^3 = (a - b)(a^2 + ab + b^2 ) is the factorization of a
difference-of-squares polynomial.

Example: Factor the binomial 8x^3 − 1.
This binomial has 8x^3, a third-degree term, as its highest- powered monomial and only a
constant, 1, is subtracted from it. We can factor by using the difference of cubes formula :

Example: Factor the polynomial b^3 – 64.
The highest degree in this binomial is 3, on the b^3 term. The only other monomial in the
binomial is 64, so the difference of cubes form is useful for factoring:

Sum of Cubes
A sum of cubes polynomial is similar to the difference of cubes polynomial, except that the
constant term is added to the third -order monomial instead of subtracted. The formula for the
factoring pattern is a^3 + b^3 = (a + b)(a^2 - ab + b^2 ) .

Example: Factor the binomial a^3 + 125.
This binomial is a cubic monomial (a^3) plus a constant (125) so we can use the formula:

Example: Factor x^3 + 27y^3.
This binomial is a cubic monomial plus another cubic monomial, so we can use the formula:

Perfect Square Trinomials
There are two patterns for the perfect square trinomial: x^2 + 2xy + y^2 = (x + y)^2 and
x^2 - 2xy + y^2 = (x - y)^2 . This trinomial has three terms: two of the terms (usually the first and
last) are monomials that we can easily take the square root of , and a third term is twice the roots
of the other two terms multiplied together. Note that only the sign of the middle term can be
positive or negative , and it matches the sign in the middle of the factored form. All of the other
coefficients must be positive.

Example: Factor x^2 – 4x + 4
This is an example of a perfect square trinomial: it is easy to take the square root of the first
and last terms (x^2 and 4), and the middle term is twice the roots of the end terms. We can use the
formula. Since the middle term is negative, we will subtract the two terms in the formula:

Example: Factor the trinomial 25x^2 + 80x + 64

1. Factor 4x^2 – 16.
Difference of squares
4x^2 – 16 = (2x + 4)(2x – 4)

2. Factor 81x^2 – 49.
Difference of squares
81x^2 – 49 = (9x + 7)(9x – 7)

3. Factor 100a^2 – 16b^2
Difference of squares
100a^2 – 16b^2 = (10a + 4b)(10a – 4b)

4. 125y^3 – 64
Difference of cubes
125y^3 – 64 = (5y – 4)(25y^2 + 20y + 16)

5. b^3 – 8a^3
Difference of cubes
b^3 – 8a^3 = (b – 2a)(b^2 + 2ab + 4a^2)

6. a^3 – 27y^3
Difference of cubes
a^3 – 27y^3 = (a – 3y)(a^2 + 3ay + 9y^2)

7. 64x^3 + 27
Sum of cubes
64x^3 + 27 = (4x + 3)(16x^2 – 12x + 9)

8. 125z^3 + 1
Sum of cubes
125z^3 + 1 = (5z + 1)(25z^2 – 5z + 1)

9. 4x^2 + 36x + 81
Perfect square trinomial
4x^2 + 36x + 81 = (2x + 9)^2

10. x^2 + 2x + 1
Perfect square trinomial
x^2 + 2x + 1 = (x + 1)^2

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