Algebra 1 Glencoe McGraw-Hill JoAnn Evans Math 8H 8-5 Factoring Differences of Squares Algebra 1 Glencoe McGraw-Hill JoAnn Evans

(x – 5) (x + 5) (x + 7) (x - 7) (3a – b) (3a + b) x2 - 25 x2 - 49 Do you remember this slide from lesson 7-7? Use FOIL to find the products. (x – 5) (x + 5) (x + 7) (x - 7) (3a – b) (3a + b) x2 - 25 x2 - 49 9a2 – b2 (y + 11) (y - 11) (2n – 5) (2n + 5) (x – 3) (x + 3) y2 - 121 4n2 - 25 x2 - 9 When the sum and difference of two terms are multiplied together, the two middle terms are opposites and will cancel out, leaving the first and last terms. The remaining terms will be squares.

When factoring, look for binomials that are difference of two squares. If you recognize that pattern you can quickly factor the expression into the sum and difference pattern. ( + ) ( - )

This expression is a binomial that is the difference of squares. 2 x2 - 4 x2 is the square of x. 4 is the square of 2. This expression is a binomial that is the difference of squares. It will factor into the sum and difference pattern: (x + 2)(x – 2) Check by multiplying.

This expression is a binomial that is the difference of squares. 5 7y 25 - 49y2 25 is the square of 5. 49y2 is the square of 7y. This expression is a binomial that is the difference of squares. Check by multiplying. It will factor into the sum and difference pattern: (5 + 7y)(5 – 7y)

x 3 x2 + 9 x2 is the square of x. 9 is the square of 3. However, this expression is a binomial that is the SUM of squares. The sum of squares doesn’t factor into the sum and difference pattern. In fact, this binomial is prime. It can’t be factored.

The expression is a binomial that is the difference of squares. 9x2 is the square of 3x. 16 is the square of 4. The expression is a binomial that is the difference of squares. 3x 4 9x2 - 16 (3x + 4)(3x – 4) n4 is the square of n2. 36 is the square of 6. The expression is a binomial that is the difference of squares. n2 6 n4 - 36 (n2 + 6)(n2 – 6)

x2 is the square of x. 14 isn’t a perfect square. This binomial can’t be factored as the difference of squares. It is prime. x x2 - 14 n8 is the square of n4. (n4)2 = n8 121 is the square of 11. The expression is a binomial that is the difference of squares. n4 11 n8 - 121 (n4 + 11)(n4 – 11)

x2 4 x4 is the square of x2. 16 is the square of 4. x4 - 16 x 2 (x2 + 4)(x2 – 4) Stop! Look again. The binomial x2 – 4 is also the difference of squares. We must FACTOR AGAIN! (x2 + 4)(x + 2)(x – 2) This is the fully factored answer.

x12 is the square of x6. 81 is the square of 9. The expression is the difference of squares. x6 9 x12- 81 x3 3 (x6 + 9)(x6 – 9) x6 – 9 is the difference of squares. We must FACTOR AGAIN! (x6 + 9)(x3 + 3)(x3 – 3) This is the fully factored answer.

48a3- 12a This expression isn’t the difference of squares. Can it be factored? Yes, first factor out the GCF. 2a 1 12a(4a2 – 1) FACTOR AGAIN! 12a(2a + 1)(2a – 1) This is the fully factored answer.

This is the fully factored answer. 5x4 – 5y4 Factor out the GCF. x2 y2 5(x4 – y4) x y 5(x2 + y2)(x2 – y2) 5(x2 + y2)(x + y)(x – y) This is the fully factored answer.

Apply Multiple Factoring Skills: 2x3 + 6x2 - 8x - 24 Factor out the GCF. 2(x3 + 3x2 – 4x – 12) Factor by grouping. 2[x2(x + 3) - 4(x + 3)] 2(x2 – 4)(x + 3) Factor the difference of squares. 2(x + 2)(x – 2)(x + 3)

Solve Equations by Factoring: 18a3 = 50a Rewrite so one side = 0. 18a3 - 50a = 0 Factor out a GCF. 2a (9a2 – 25) = 0 difference of squares 2a (3a + 5)(3a – 5) = 0 Zero Product Property 2a = 0 3a + 5 = 0 3a – 5 = 0 a = 0,

Solve Equations by Factoring: Rewrite so one side = 0. difference of squares Zero Product Property