Think about it... In a game, you see two cards, a 5 and a 6. You are then dealt two more cards with numbers on them. All the cards in the deck are positive. You win if... The PRODUCT of your two numbers equals the 6 card shown, AND... The SUM of your two numbers equals the 5 card shown. What should your two cards be for you to win the game? Is there more than one answer? Explain. 4-4 Factoring Quadratic Expressions To find common and binomial factors of quadratic expressions. To factor special quadratic expressions.

Factoring ▪ Factors are numbers that have a product equal to (multiply to) a given number. For example, factors of 6 are 2 and 3. Factors of a given expression are expressions that have a product equal to the given expression. Factoring is rewriting an expression as a product of its factors. Basically we are “undoing” FOIL. ▪ Essential Understanding - You can factor many quadratic trinomials (ax 2 + bx + c) into products of two binomials. ▪ You can use the Distributive Property or the undoing of F O I L method to factor into two binomials. ▪ Let’s remind ourselves what F O I L is...

▪ OK, let’s try to FOIL in reverse now. Find two numbers who multiply to be the “c” part and then sum to be the “b” part. ▪ Let’s un-FOIL this... x 2 + 4x + 4 Problem 1Factoring ax 2 + bx + c when a = ± 1 (a)x 2 + 9x + 20(b)x x – 72(c)-x x - 12

Problem 1Got it?Factoring ax 2 + bx + c when a = ± 1 (a)x x + 40(b)x x + 30(c)-x x + 32

Warm Up Factoring ax 2 + bx + c when a = ± 1 (a)x x + 49(b)x 2 + 8x + 15(c)-x 2 + 4x + 45

Warm Up!! Factor these trinomials!! x x + 48x 2 – x – 20-x 2 – 3x + 40

▪ The Greatest Common Factor of an expression is a common factor of the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. You can factor any expressions that have a GCF not equal to 1. Problem 2Finding Common Factors What is the expression in factored form? (a)6x 2 + 9x(b)4x x - 56

Problem 2Got it?Finding Common Factors (a)7x (b)9x 2 + 9x - 18(c)4x 2 + 8x + 12 Nothing multiplies to be 3 and combines to be 2 so this cannot be factored any further!

Warm Up Place on a separate piece of paper. Factor (a)x x + 48 Factor (b) -x 2 - x + 12 GCF (c)9x 2 + 6x GCF then Factor (d)9x x + 27

▪To factor a quadratic with a ≠ 1, and there is no common factor, we have to start a little differently. Problem 3Factoring ax 2 + bx + c when a ≠ 1 (a)2x x + 12(b)4x 2 - 4x - 3 Create diamond a c b factor Make a Box 3x 8x 2x 2 12 Remove GCF + 3 2x x + 4 Create diamond a c b factor Make a Box -6x 2x 4x 2 -3 Remove GCF - 3 2x

Problem 3Got it?Factoring ax 2 + bx + c when a ≠ 1 (a)4x 2 + 7x + 3(b)2x 2 - 7x + 6 Think about it...Can you factor 2x 2 + 2x + 2 into the product of two binomials? 12 4 Nothing multiplies to be 4 and combines to be 2 so this cannot be factored! 3x 4x 4x x x x -4x 2x x x - 2

Warm UP Put on a separate piece of paper please. Factoring ax 2 + bx + c when a ≠ 1 (a)2x 2 + 9x + 10(b)3x 2 + 8x x 4x 2x x x + 2 9x -x 3x x 3x

Special Factor Pairs ▪ A perfect square trinomial is a trinomial that is the square of a binomial. Problem 4Factoring Perfect Square Trinomials What is 4x x + 25 in factored form? Got it? What is 64x x + 1 in factored form?

Special Factor Pairs ▪ The expression a 2 – b 2 is the difference of two squares. There is a pattern to its factors. ▪ Can you spot it? Problem 5Factoring a Difference of Two Squares What is x in factored form?Got it? What is x in factored form?

Problem 5Factoring a Difference of Two Squares What is 25x in factored form? Got it? What is 16x in factored form?