# 3.3 – Special Cases

## Objectives

• Identify and factor a difference of two squares.
• Identify and factor a perfect square trinomial.
• Expand binomials to produce a difference of two squares.

## Key Terms

• Difference of Squares – An expression that contains two perfect squares with one subtracted from the other.
• A difference of squares can always be factored as follows: $a^2-b^2 = (a + b)(a - b)$
• Difference of Two Cubes – An expression that contains two perfect cubes with one subtracted from the other.
• A difference of two cubes can always be factored as follows: $a^3-b^3=(a-b)(a^2+ab+b^2)$
• Perfect Square Trinomial – The result of multiplying a binomial by itself.
• In other words, it is the square of a polynomial with two terms.
• Sum of Two Cubes – An expression that contains that contains two perfect cubes, with one added to other other.
• This type of expression can always be factored as follows: $a^3+b^3=(a+b)(a^2-ab+b^2)$
• Example: $8x^3+y^6$
• $8x^3=2x\bullet 2x\bullet 2x$
• $y^6=y^2\bullet y^2\bullet y^2$
• So, $(2x+y^2)(4x^2-2xy^2+y^4)$

## Notes

Difference of Squares
• Properties of the Difference of Squares
• All variables are raised to an even power
• There are only two terms
• Both terms have negative coefficients
• Both coefficients are perfect squares
• There are only two terms
• One term has a negative coefficient and one term has a positive coefficient
• The middle term of the difference of squares is 0x
• 0 times x = 0

Perfect Square Trinomials
• Properties of Perfect Square Trinomials
• The first and third terms must be perfect squares.
• Neither of the perfect squares can have a minus sign
• If the perfect square terms are $A^2$ and $B^2$ then the other term must be 2AB.

Difference of Two Cubes
• Formulas and Examples

Examples
•  Example 1:  Difference of Squares
• When asked to factor the expression $x^2-121$, a student gives the answer (x-11)(x-11).
• What is the wrong with this answer?
• Answer:  One of the minus signs should be a plus sign.
• Example 2:  Difference of Squares
• To factor $9x^2-16$, you can first rewrite the expression as: $(3x)^2-(4)^2$.
• Example 3:  Difference of Squares
• What is the factorization of $16x^2 - 49$?
• Answer: (4x + 7)(4x – 7)
• Example 4:  Difference of Squares
• What is the factorization of $164A^2 - 81B^2$?
• Answer: (8A + 9B)(8A – 9B)
• Example 5:  Difference of Squares
• In the polynomial below, what number should replace the question mark to produce a difference of squares?
• $x^2 + ?x - 49$
• Example 6:  Difference of Squares
• What is the product of the two binomials below?
• (4A + 5B)(4A – 5B)
• Answer: $16A^2 - 25B^2$
• Example 7:  Sum of Two Cubes
• Factor the expression: $343x^3 + 216y^3$
• Answer: $(7x + 6y)(49x^2 - 42xy + 36y^2)$
• Example 8: Difference of Two Cubes
• What is the factorization of $2x^2 - 16x + 32$?
• Answer: $2(x - 4)^2$
• Example 9:  Perfect Square Trinomials
• What is the factorization of $121x^2 - 22x + 1$?
• Answer: $(11x - 1)^2$
• Example 10:  Perfect Square Trinomials
• What value in place of the question mark makes the polynomial below a perfect square trinomial? \$16x^2 + ?x + 36\$\$