Review Factoring Trinomials in the form ax2 + bx + c, (a = 1) FACTORING REVERSES MULTIPLICATION. For example: What can you multiply to get x2? Answer: x times x Therefore, x2 in factored form is x(x). List pairs of factors of 12. 1 & 12, 2 & 6, 3 & 4 Objective: To factor trinomials. Steps for Factoring x2 + bx + c Watch and think. Factor x2 – 9x + 14 . 1. 2. 3. 4. 5. 6. Factor out GCF. Find factors of “c” that combine to give you “b”. When “c” is positive, the sign in each binomial is the same as “b”. If “c” is negative, the sign in each binomial is different. Write as the product of 2 binomials. GCF( )( ) Use FOIL or the Distributive Prop. to check. Examples with positive “c” term a) g2 + 7g + 10 b) v2 + 21v + 20 c) a2 + 13a + 30 d) q2 – 15q + 36 Check for GCF. (There is no GCF.) For the first term, x2 = (x )(x ) Because “c” is positive, add factors of “c” to get “b”, which is 9. Factors of 14 are 1 & 14, 2 & 7. Which pair will combine to = 9? (x 2) (x 7) Now the signs: The last term is positive which means the signs are the same. Look at the middle term. The signs need to be negative. Final answer: (x - 2) (x - 7) Check with FOIL: x2 – 7x – 2x + 14 which equals x2 – 9x + 14, the original problem. Examples – Solutions (positive “c” means signs will be same) a) g2 + 7g + 10 Factors of 10: 1x10, 2x5 2 + 5 = 7 so factored form = (g + 2)(g + 5) b) v2 + 21v + 20 Factors of 20: 1x20, 2x10, 4x5 1 + 20 = 21 so factored form = (v + 1)(v + 20) c) a2 + 13a + 30 Factors of 30: 1x30, 2x15, 3x10, 5x6 3 + 10 = 13 so factored form = (a + 3)(a + 10) d) q2 – 15q + 36 Factors of 36: 1x36, 2x18, 3x12, 4x9 3 + 10 = 13 so factored form = (q + 12)(q + 3) 1 Examples Examples -- Solutions e) b2 + 11b + 10 e) b2 + 11b + 10 = (b + 10)(b + 1) f) k2 – 10k + 25 f) k2 – 10k + 25 = (k – 5)(k – 5) g) x2 – 11xy + 18y2 g) x2 – 11xy + 18y2 = (x – 2y)(x - 9y) h) n2 – 9mn + 20n2 h) n2 – 9mn + 20n2 = (n - 4m)(n – 5m) Watch and think. Factor 3x3 – 3x2 – 18x . What if “c” is negative? Every example we’ve done so far has had a positive “c” term. Therefore, we’ve combined the factors of “c” to get the “b” term and both terms had same sign. Now we’ll look at when “c” is negative. What do you think will change in how we pick the factors to use?...Instead of terms having the same sign, the terms will have different signs. Examples with negative “c” term a) m2 + 8m – 20 b) p2 – 3p – 40 c) y2 – y – 56 d) n2 – 5n - 24 Check for GCF. (GCF = 3x.) Factor it out: 3x(x2 – x – 6). Factor the first term in the trinomial, 3x(x )(x ) Because “c” is negative, one factor will be positive and one will be negative. (Think of subtracting the factors since signs are different.) Factors of 6 are 1 & 6 and 2 & 3. Which pair will subtract to give 1? 3x(x 2) (x 3) Now the signs: The last term is negative which means the signs are different. Look at the middle term. Which number needs to be negative to give -1? Final answer: 3x(x + 2) (x - 3) Check with FOIL: 3x(x2 – 3x + 2x – 6) which gives 2 3x(x –x- 6), which equals 3x3 – 3x2 – 18x, the original problem. Examples -- Solutions a) m2 + 8m – 20 = (m + 10)(m – 2) Factors of 20: 1x20, 2x10, 4x5; 10 and -2 combine to = 8 (which is “b”) b) p2 – 3p – 40 = (p – 8)( p + 5) Factors of 40: 1x40, 2x20, 4x10, 5x8; -8 and 5 combine to = -3 c) y2 – y – 56 = (y – 8)(y + 7) Factors of 56: 1x56, 2x28, 4x14, 7x8; -8 and 7 combine to = -1 d) n2 – 5n – 24 = (n – 8)(n + 3) 2 Examples x2 + 11xy + Examples 24y2 e) f) v2 + 2vw – 48w2 g) m2 – 17mn – 60n2 h) d2 + 17dg – 60g2 Note: If there is a variable squared at the end of the trinomial, simply add that variable to the end of each term in factored form. e) x2 + 11xy + 24y2 = (x + 3y)(x + 8y) Factors of 24: 1x24, 2x12, 3x8, 4x6; 3 and 8 combine to = 11 (which is “b”) f) v2 + 2vw – 48w2 = (v - 6w)(v + 8w) Factors of 48: 1x48, 2x24, 3x16, 4x12, 6x8; -6 and 8 combine to = 2 g) m2 – 17mn – 60n2 = (m - 20n)(m + 3n) Factors of 60: 1x60, 2x30, 3x20, 4x15, 6x10; -20 and 3 combine to = -17 h) d2 + 17dg – 60g2 = (d + 20g)(d - 3g) SUMMARY Factoring x2 + bx + c Find factors of “c” that combine to give you “b”. 2. If “c” is positive, the terms have the same sign. 3. If “c” is negative, the terms will have different signs. 4. Use FOIL to check. 1. 3
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