1. Essential Question: How is FOIL related to factoring?

2. 5-4: Factoring Quadratic Expressions Quadratic Functions (you saw this in 5-1) A quadratic function is one whose largest term uses x 2 It’s written in standard form as f(x) = ax 2 + bx + c a, b, and c represent coefficients (real numbers) The x 2 terms comes first, followed by the x term, followed by the term that doesn’t have an x The x 2 term and x term cannot be combined

3. 5-4: Factoring Quadratic Expressions FOIL (Note: You saw this in 5-1) FOIL is an acronym for “First, Outer, Inner, Last” Multiply the indicated terms together Combine like terms Example: y = (2x + 3)(x – 4) y = (2x + 3)(x – 4) First Last Inner Outer First: 2x x = 2x 2 Outer: 2x -4 = -8x Inner: 3 x = 3x Last: 3 -4 = -12 y = 2x 2 – 8x + 3x – 12 y = 2x 2 – 5x - 12

4. 5-4: Factoring Quadratic Expressions FOIL (x – 4)(x + 3) (-x – 5)(3x – 1) x 2 – 4x + 3x – 12 x 2 – x – 12 -3x 2 – 15x + x + 5 -3x 2 – 14x + 5

5. 5-4: Factoring Quadratic Expressions Finding the Greatest Common Factor (GCF) The GCF of an expression is the common factor with the greatest coefficient and the smallest exponent Example: Factor 4x 6 + 20x 3 – 12x 2 The largest coefficient that can divide 4, 20, and -12 is 4 The smallest exponent is x 2 4x 2 (x 4 ) + 4x 2 (5x) + 4x 2 (-3) 4x 2 (x 4 + 5x – 3)

6. 5-4: Factoring Quadratic Expressions Factor 4w 2 + 2w 5t 4 + 7t 2 GCF: 2w Factored: 2w(2w + 1) GCF: 1t 2 Factored: t 2 (5t 2 + 7)

7. 5-4: Factoring Quadratic Expressions Assignment FOIL/GCF worksheet Do all problems Show your work

8. Essential Question: How is FOIL related to factoring?

9. 5-4: Factoring Quadratic Expressions Factoring: The steps (Holy Grail algorithm) In standard form: f(x) = ax 2 + bx + c 1. Find two numbers with: A product of a c A sum of b 2. Use those two numbers to split the “b” term 3. Factor out the GCF from the first two terms as well as the last two terms 4. You know you’ve factored correctly if both binomials inside the parenthesis match 5. Combine the terms outside parenthesis into their own parenthesis +

10. 5-4: Factoring Quadratic Expressions Some hints (summarized on next slide): The a term should be positive (I won’t give you otherwise) If not, flip the signs on each term -x 2 + 5x + 24 gets flipped into x 2 – 5x – 24 If a c is positive, the two numbers you’re looking for are going to be the same sign as b ex #1) x 2 + 9x + 204 & 5 ex #2) x 2 – 11x + 28-4 & -7 Why? Because only a positive positive and/or negative negative = positive If a c is negative, the bigger of the two numbers will have the same sign as b ex #3) x 2 + 3x – 105 & -2 ex #4) x 2 – 5x – 24-8 & 3 Why? Because only a negative positive = negative

11. 5-4: Factoring Quadratic Expressions Multiply: + number Multiply: - number Add: + number Add: - number Add: + number Add: - number Both #s are +Both #s are -Bigger # is +Bigger # is - Some hints about finding the two numbers to be used in factoring:

12. 5-4: Factoring Quadratic Expressions Factoring (Example #4) Factor: 3x 2 – 16x + 5 a = 3, c = 5 → ac = 15 Find two numbers that: multiply together to get 15 add to get -16 Possibilities: -1/-15, -3/-5 Rewrite the b term 3x 2 – 1x – 15x + 5 Factor GCF from first two and last two terms x(3x – 1) – 5(3x – 1) Combine terms outside the parenthesis (x – 5)(3x – 1) 3x 2 – 16x + 5 + -153x 2 x x + 5 x(3x – 1)-5(3x – 1) (x – 5)(3x – 1)

13. 5-4: Factoring Quadratic Expressions Factor 2x 2 + 11x + 12

14. 5-4: Factoring Quadratic Expressions Factoring (Example #5) Factor: 4x 2 – 4x – 15 a = 4, c = -15 → ac = -60 Find two numbers that: multiply together to get -60 (1 positive, 1 negative) add to get -4 (larger is negative) Possibilities: 1/-60, 2/-30, 3/-20, 4/-15, 5/-12, 6/-10 Rewrite the b term 4x 2 + 6x – 10x – 15 Factor GCF from first two and last two terms 2x(2x + 3) – 5(2x + 3) Combine terms outside the parenthesis (2x – 5)(2x + 3) 4x 2 – 4x – 15 + +6 -10 4x 2 x x – 15 2x(2x + 3)-5(2x + 3) (2x – 5)(2x + 3)

15. 5-4: Factoring Quadratic Expressions Factor 6x 2 + 11x – 35

16. 5-4: Factoring Quadratic Expressions Assignment Pg. 263 25 – 36 (all problems) No work = no credit Additional examples (and steps) are available at http://www.gushue.com/factoring2.phphttp://www.gushue.com/factoring2.php

17. Essential Question: How is FOIL related to factoring?

18. 5-4: Factoring Quadratic Expressions Factoring: The steps (same as last week) In standard form: f(x) = ax 2 + bx + c Find two numbers with: A product of a c A sum of b Use those two numbers to split the “b” term Factor out the GCF from the first two terms as well as the last two terms You know you’ve factored correctly if both binomials inside the parenthesis match Combine the terms outside parenthesis into their own parenthesis

19. 5-4: Factoring Quadratic Expressions Factoring (Example #1) Factor: x 2 + 8x + 7 a = 1, c = 7 → ac = 7 Find two numbers that: multiply together to get 7 add to get 8 Only possibility is 1/7 Rewrite the b term x 2 + 1x + 7x + 7 Factor GCF from first two and last two terms x(x + 1) + 7(x + 1) Combine terms outside the parenthesis (x + 7)(x + 1) x 2 + 8x + 7 + +1 +7 x 2 x x + 7 x(x + 1)+7(x + 1) (x + 7)(x + 1)

20. 5-4: Factoring Quadratic Expressions Your Turn. Factor: x 2 + 4x – 5 x 2 – 12x + 11 Two numbers? 5 & -1 x 2 + 5x – 1x – 5 x(x + 5) -1(x + 5) (x – 1)(x + 5) Two numbers? -11 & -1 x 2 – 11x – 1x + 11 x(x – 11) -1(x – 11) (x – 1)(x – 11)

21. 5-4: Factoring Quadratic Expressions Multiply: + number Multiply: - number Add: + number Add: - number Add: + number Add: - number Both #s are +Both #s are -Bigger # is +Bigger # is - Some hints about finding the two numbers to be used in factoring:

22. 5-4: Factoring Quadratic Expressions Factoring (Example #2) Factor: x 2 – 17x + 72 a = 1, c = 72 → ac = 72 Find two numbers that: multiply together to get 72 (both + or both –) add to get -17 (both –) Possibilities: -1/-72, -2/-36, -3/-24, -4/-18, -6/-12, -8/-9 Rewrite the b term x 2 – 8x – 9x + 72 Factor GCF from first two and last two terms x(x – 8) + -9(x – 8) Combine terms outside the parenthesis (x – 9)(x – 8)

23. 5-4: Factoring Quadratic Expressions Your Turn. Factor: x 2 + 8x + 15 x 2 – 5x + 6 Two numbers? -2 & -3 x 2 – 2x – 3x + 6 x(x – 2) -3(x – 2) (x – 3)(x – 2) Two numbers? 3 & 5 x 2 + 3x + 5x + 15 x(x + 3) +5(x + 3) (x + 5)(x + 3)

24. 5-4: Factoring Quadratic Expressions Factoring (Example #3) Factor: x 2 – x – 12 a = 1, c = -12 → ac = -12 Find two numbers that: multiply together to get -12 (one + & one –) add to get -1 (bigger number is –) Possibilities: -1/12, -12/1, -2/6, -6/2, -3/4, -4/3 Rewrite the b term x 2 – 4x + 3x – 12 Factor GCF from first two and last two terms x(x – 4) + 3(x – 4) Combine terms outside the parenthesis (x + 3)(x – 4)

25. 5-4: Factoring Quadratic Expressions Your Turn. Factor: x 2 + 4x – 12 x 2 – 2x – 15 Two numbers? -2 & 6 x 2 – 2x + 6x – 12 x(x – 2) +6(x – 2) (x + 6)(x – 2) Two numbers? 3 & -5 x 2 + 3x – 5x – 15 x(x + 3) -5(x + 3) (x – 5)(x + 3)

26. 5-4: Factoring Quadratic Expressions Assignment Pg. 263 7 – 24 (all problems) Additional examples (and steps) are available at http://www.gushue.com/factoring.phphttp://www.gushue.com/factoring.php

27. Essential Question: How is FOIL related to factoring?

28. 5-4: Factoring Quadratic Expressions There are two special cases to discuss: The Difference of Perfect Squares x 2 – 16 If we’re using the Holy Grail Algorithm: a = 1 b = 0 (there’s no ‘x’ term) c = -16 So we’re looking for two numbers that multiply to get -16 (1 -16) and add together to get 0 The only way to have two numbers that add together to get 0 is if they’re opposites, in this case 4 & -4

29. 5-4: Factoring Quadratic Expressions Factoring: x 2 - 16 x 2 + 0x – 16 + -4+4 x 2 x x – 16 x(x – 4)+4(x – 4) (x + 4)(x – 4)

30. 5-4: Factoring Quadratic Expressions Factor 9x 2 – 25 The shortcut: Take the square root of the left term: Take the square root of the right term: Write the factor as a sum and difference of the squares 3x 5 (3x + 5)(3x – 5)

31. 5-4: Factoring Quadratic Expressions Perfect Square Trinomial x 2 + 6x + 9 If we’re using the Holy Grail Algorithm: a = 1 b = 6 c = 9 So we’re looking for two numbers that multiply to get 9 (1 9) and add together to get 6 Those numbers have to be 3 & 3 A perfect square trinomial occurs when the numbers are the same.

32. 5-4: Factoring Quadratic Expressions Factoring: x 2 + 6x + 9 x 2 + 6x + 9 + +3 x 2 x x + 9 x(x + 3)+3(x + 3) (x + 3)(x + 3) written as (x + 3) 2

33. 5-4: Factoring Quadratic Expressions Factor 16x 2 – 56x + 49 The shortcut: Take the square root of the left term: Take the square root of the right term: The sign both terms share will be the sign of the middle term: 4x 7 (4x – 7)(4x – 7) = (4x – 7) 2

34. 5-4: Factoring Quadratic Expressions Assignment Pg. 264 37 – 45 (all problems) No work = no credit