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MTH 10905 Algebra

MTH 10905 Algebra. Factoring Trinomials of the form ax 2 + bx + c where a = 1 Chapter 5 Section 3. Factoring Trinomials. It is important that you understand sections 5.3 and 5.4 to be successful in Chapter 6.

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MTH 10905 Algebra

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  1. MTH 10905Algebra Factoring Trinomials of the form ax2 + bx + c where a = 1 Chapter 5 Section 3

  2. Factoring Trinomials • It is important that you understand sections 5.3 and 5.4 to be successful in Chapter 6. • In this section we learn factoring a trinomial in the form of ax2 + bx + c where a = 1; for example, x2 + 5x + 6 • In section 5.4 we learn factoring a trinomial in the form of ax2 + bx + c where a ¹ 1; for example, 2x2 + 7x + 3

  3. Trial and Error • When factoring x2 + bx + c you will always get (x + ) (x + ) • Write down the factors of the constant, c, and try them in the shaded area. • You need the factors of c that sum to b • Use the FOIL method to check

  4. Trinomials in the form ax2 + bx + c, a=1 • It is important that you know what a, b and c equals. • Examples: • x2 + 7x + 12 = (x + 3)(x + 4) FOIL a = 1, b = 7, c = 12 (3)(4) = 12 factors of 12 that sum to 7 3 + 4 = 7 • x2 – 2x – 24 = (x – 6)(x + 4) FOIL a = 1, b = -2, c = -24 (-6)(4) = -24 factors of -24 that sum to -2 (-6)+(4) = -2

  5. Trial and Error Example: x2 + 12x + 20 a=1 b=12 c=20 (x + 2)(x + 10) You can always check using FOIL method (x + 2) (x + 10) (x)(x) + (x)(10)+ (2)(x) + (2)(10) x2 + 10x + 2x + 20 x2 + 12x + 20

  6. Helpful Hint • If b = neg, c = pos, then factor = 2 negative • If b = neg, c = neg, then factor = 1 pos 1 neg • If b = pos, c = neg, then factor = 1 pos 1 neg • If b = pos, c = pos, then factor = 2 positive Example: Using x2 + bx – c , determine the sign of the numbers in the factors: One positive and one negative factor

  7. Factoring Trinomials Example: x2 + x – 72 a=1 b=1 c=-72 (x + 9)(x – 8) Check using FOIL (x + 9) (x – 8) (x)(x) + (x)(-8)+ (9)(x) + (9)(-8) x2 + (-8x) + 9x + (-72) x2 + x – 72

  8. Factoring Trinomials Example: x2 – x – 72 a=1 b=-1 c=-72 (x – 9)(x + 8) Check using FOIL (x – 9)(x + (8) (x)(x) + (x)(8)+ (-9)(x) + (-9)(8) x2 + 8x + (-9x) + (-72) x2 – x – 72

  9. Factoring Trinomials Example: x2 – 11x + 30 a=1 b=-11 c=32 (x – 5)(x – 6) Check using FOIL (x – 5)(x – 6) (x)(x) + (x)(-6)+ (-5)(x) + (-5)(-6) x2 + (-6x) + (-5x) + 30 x2 – 11x + 30

  10. Factoring Trinomials Example: t2 + 4x – 32 a=1 b=4 c=-32 (t – 4)(t + 8) Check using FOIL (t – 4)(t + 8) (t)(t) + (t)(8)+ (-4)(t) + (-4)(8) t2 + (8t) + (-4t) – 32 t2 + 4t – 32

  11. Factoring Trinomials Example: x2 – 14x + 49 a=1 b=-14 c=49 (x – 7)(x – 7) (x – 7)2 Check using FOIL (x – 7)(x – 7) (x)(x) + (x)(-7)+ (-7)(x) + (-7)(-7) x2 + (-14x) + 49 x2 – 14x + 49

  12. Factoring Trinomials Example: x2 – 2x – 63 a=1 b=-2 c=-63 (x – 9)(x + 7) Check using FOIL (x – 9)(x + 7) (x)(x) + (x)(7)+ (-9)(x) + (-9)(7) x2 + 7x + (-9x) – 63 x2 – 2x – 63

  13. Factoring Trinomials Example: x2 + 10x + 20 a=1 b=10 c=20 PRIME A polynomial that cannot be factored using only integer coefficients is called a prime polynomial. No factors of c that can sum to b.

  14. Factoring Trinomials Example: x2 + 4xy + 4y2 a=1 b=4 c=4 (x + 2y)(x + 2y) (x + 2y)2 Check using FOIL (x + 2y)(x + 2y) (x)(x) + (x)(2y)+ (2y)(x) + (2y)(2y) x2 + 2xy + 2xy + 4y2 x2 + 4xy + 4y2 The last term of the factors must contain a y in order to get the y2

  15. Factoring Trinomials Example: x2 – xy – 30y2 a=1 b=1 c=-30 (x – 6y)(x + 5y) The last term of the factors must contain a y in order to get the y2 Check using FOIL (x – 6) (x + 5y) (x)(x) + (x)(5y)+ (-6y)(x) + (-6y)(5y) x2 + 5xy + (-6xy) + (-30y2) x2 – xy – 30y2

  16. Removing a Common Factorfrom a Trinomials Example: 3x2 – 21x + 18 3(x2 – 7x + 6) 3(x – 1)(x – 6) Sometimes each term has a GCF that we must pull out first making it easier to factor. Check using FOIL 3(x – 1)(x – 6) 3[(x)(x) + (x)(-6)+ (-1)(x) + (-1)(-6)] 3[x2 + (-6x) + (-1x) + 6] 3[x2 – 7x + 6] 3x2 – 21x + 18

  17. Removing a Common Factorfrom a Trinomials Example: 3m3 + 9m2 – 84m 3m(m2 + 3m – 28) 3m(m + 7)(m – 4) Check using FOIL 3m(m + 7)(m – 4) 3m[(m)(m) + (m)(7)+ (7)(m) + (7)(-4)] 3m[m2 + 7m + 7m + -28] 3m[m2 + 14m - 28] 3m3 + 9m2 – 84

  18. REMEMBER • Always factor out the GCF first. • A table can be helpful. Use one column for all possible factors of c an another column for the sum of the factors. • One or both factors of c can be negative. • When c is positive, the two factors will have the same sign as b. • When c is negative, the two factors will have opposite signs. • When the factors have opposite signs, the larger of the two will be the same sign as b • You should always check your work by multiplying.

  19. HOMEWORK 5.3 Page 311: #17, 19, 21, 27, 39, 45, 63, 73

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