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Unit #3: Quadratics 5-4: Factoring Quadratic Expressions

Unit #3: Quadratics 5-4: Factoring Quadratic Expressions. Essential Question: How is FOIL related to factoring?. 5-4: Factoring Quadratic Expressions. Quadratic Functions (you saw this in 5-1) A quadratic function is one whose largest term uses x 2

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Unit #3: Quadratics 5-4: Factoring Quadratic Expressions

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  1. Unit #3: Quadratics5-4: Factoring Quadratic Expressions 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 x2 • It’s written in standard form asf(x) = ax2 + bx + c • a, b, and c represent coefficients (real numbers) • The x2 terms comes first, followed by the x term, followed by the term that doesn’t have an x • The x2 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) Last First First: 2x • x = 2x2 Outer: 2x • -4 = -8x Inner: 3 • x = 3x Last: 3 • -4 = -12 y = 2x2– 8x + 3x – 12 y = 2x2 – 5x - 12 Inner Outer

  4. 5-4: Factoring Quadratic Expressions • FOIL • (x – 4)(x + 3) • (-x – 5)(3x – 1) x2 – 4x + 3x – 12 x2 – x – 12 -3x2 – 15x + x + 5 -3x2 – 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 4x6 + 20x3 – 12x2 • The largest coefficient that can divide 4, 20, and -12 is 4 • The smallest exponent is x2 • 4x2(x4) + 4x2(5x) + 4x2(-3) • 4x2(x4 + 5x – 3)

  6. 5-4: Factoring Quadratic Expressions • Factor • 4w2 + 2w • 5t4 + 7t2 GCF: 2w Factored: 2w(2w + 1) GCF: 1t2 Factored: t2(5t2 + 7)

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

  8. Unit #3: Quadratics5-4: Factoring Quadratic ExpressionsDay 2 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) = ax2 + 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 +

  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 • -x2 + 5x + 24 gets flipped into x2 – 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) x2 + 9x + 20 4 & 5 • ex #2) x2 – 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) x2 + 3x – 10 5 & -2 • ex #4) x2 – 5x – 24 -8 & 3 • Why? Because only a negative • positive = negative

  11. 5-4: Factoring Quadratic Expressions Some hints about finding the two numbers to be used in factoring:

  12. 5-4: Factoring Quadratic Expressions • Factoring (Example #4) • Factor: 3x2 – 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 • 3x2 – 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) 3x2 – 16x + 5 + 3x2 x x + 5 -1 -15 x(3x – 1) -5(3x – 1) (x – 5)(3x – 1)

  13. 5-4: Factoring Quadratic Expressions • Factor • 2x2 + 11x + 12

  14. 5-4: Factoring Quadratic Expressions 4x2 – 4x – 15 • Factoring (Example #5) • Factor: 4x2 – 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 • 4x2+ 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) + 4x2 x x – 15 +6 -10 2x(2x + 3) -5(2x + 3) (2x – 5)(2x + 3)

  15. 5-4: Factoring Quadratic Expressions • Factor • 6x2 + 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.php

  17. Unit #3: Quadratics5-4: Factoring Quadratic ExpressionsDay 3 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) = ax2 + 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: x2 + 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 • x2 + 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) x2 + 8x + 7 + +7 x2 x x + 7 +1 x(x + 1) +7(x + 1) (x + 7)(x + 1)

  20. 5-4: Factoring Quadratic Expressions • Your Turn. Factor: • x2 + 4x – 5 • x2 – 12x + 11 Two numbers? 5 & -1 x2+ 5x – 1x – 5 x(x + 5) -1(x + 5) (x – 1)(x + 5) Two numbers? -11 & -1 x2– 11x – 1x + 11 x(x – 11) -1(x – 11) (x – 1)(x – 11)

  21. 5-4: Factoring Quadratic Expressions Some hints about finding the two numbers to be used in factoring:

  22. 5-4: Factoring Quadratic Expressions • Factoring (Example #2) • Factor: x2 – 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 • x2 – 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: • x2 + 8x + 15 • x2 – 5x + 6 Two numbers? 3 & 5 x2+ 3x + 5x + 15 x(x + 3) +5(x + 3) (x + 5)(x + 3) Two numbers? -2 & -3 x2– 2x – 3x + 6 x(x – 2) -3(x – 2) (x – 3)(x – 2)

  24. 5-4: Factoring Quadratic Expressions • Factoring (Example #3) • Factor: x2 – 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 • x2 – 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: • x2 + 4x – 12 • x2 – 2x – 15 Two numbers? -2 & 6 x2– 2x + 6x – 12 x(x – 2) +6(x – 2) (x + 6)(x – 2) Two numbers? 3 & -5 x2+ 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.php

  27. Unit #3: Quadratics5-4: Factoring Quadratic ExpressionsDay 4 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 • x2 – 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: x2 - 16 x2 + 0x – 16 + x2 x x – 16 -4 +4 x(x – 4) +4(x – 4) (x + 4)(x – 4)

  30. 5-4: Factoring Quadratic Expressions • Factor • 9x2 – 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 • x2 + 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: x2 + 6x + 9 x2 + 6x + 9 + x2 x x + 9 +3 +3 x(x + 3) +3(x + 3) (x + 3)(x + 3) written as (x + 3)2

  33. 5-4: Factoring Quadratic Expressions • Factor • 16x2 – 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

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