Factoring. Factoring Polynomial Equations. Special Factoring Patterns. Factoring. Special Factoring Patterns. Special Factoring Patterns

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1 Factoring Factoring Polynomial Equations Ms. Laster Earlier, you learned to factor several types of quadratic expressions: General trinomial - 2x 2-5x-12 = (2x + 3)(x - 4) Perfect Square Trinomial - x 2 +10x+25=(x + 5) 2 Difference of - 4x 2-9 =(2x + 3)(2x - 3) Common monomial factor - 6x 2 +15x=3x(2x+5) Factoring Earlier, you learned to factor several types of quadratic expressions: General trinomial - 2x 2-5x-12 = (2x + 3)(x - 4) Perfect Square Trinomial - x 2 +10x+25=(x + 5) 2 Difference of - 4x 2-9 =(2x + 3)(2x - 3) Common monomial factor - 6x 2 +15x=3x(2x+5) Now, we ll look at some other types of polynomials. Sum of two cubes a 3 + b 3 = (a + b)(a 2 -ab + b 2 ) Sum of two cubes a 3 + b 3 = (a + b)(a 2 -ab + b 2 ) x = (x + 2)(x 2-2x + 4)

2 Difference of two cubes a 3 -b 3 = (a - b)(a 2 + ab + b 2 ) Difference of two cubes a 3 -b 3 = (a - b)(a 2 + ab + b 2 ) 8x 3-1 = (2x - 1)(2x 2 + 2x + 1) Factor 64a 4-27a 64a 4-27a Factor common monomial 64a 4-27a a(64a 3-27) Factor common monomial 64a 4-27a Factor common monomial a(64a 3-27) Difference of cubes

3 64a 4-27a Factor common monomial a(64a 3-27) Difference of cubes a((4a) ) 64a 4-27a Factor common monomial a(64a 3-27) Difference of cubes a((4a) ) a(4a - 3)(16a 2 +12a + 9) Sometimes, you can factor a polynomial by grouping pairs of terms that have a common monomial factor. Sometimes, you can factor a polynomial by grouping pairs of terms that have a common monomial factor. The pattern for this is: ra + rb + sa + sb = r(a + b) + s(a + b) =(r + s)(a + b) Factor x 2 y 2-3x 2-4y x 2 y 2-3x 2-4y x 2 is common to the first two terms, and 4 is common to the second two.

4 x 2 y 2-3x 2-4y x 2 (y 2-3)-4(y 2-3) factored out a -4 remember, you x 2 y 2-3x 2-4y x 2 (y 2-3) -4(y 2-3) (x 2-4)(y 2-3) 1st term is a difference of x 2 y 2-3x 2-4y x 2 (y 2-3) -4(y 2-3) (x 2-4)(y 2-3) (x - 2)(x + 2)(y 2-3) Sometimes an expression will be in quadratic form, but not obviously. Any expression in the form au 2 + bu + c, where u is some expression of x, is quadratic. Sometimes an expression will be in quadratic form, but not obviously. Any expression in the form au 2 + bu + c, where u is some expression of x, is quadratic. 81x 4-16 This is not obviously quadratic, since 81x 4 doesn t look like a perfect square, but let s look again. (9x 2 ) Now, this is a difference of

5 (9x 2 ) Now, this is a difference of (9x 2-4)(9x 2 + 4) (9x 2 ) Now, this is a difference of (9x 2-4)(9x 2 + 4) That first term is another difference of (9x 2 ) Now, this is a difference of (9x 2-4)(9x 2 + 4) (3x - 2)(3x + 2)(9x 2 + 4) Try factoring a 2 b 2-8ab b 4 Common factor b 2 (a 2-8ab + 16b 2 )

6 b 2 (a 2-8ab + 16b 2 ) Perfect Square Trinomial b 2 (a 2-8ab + 16b 2 ) b 2 (a 2-4b)(a 2-4b) or b 2 (a 2-4b) 2 Summary of Methods for Factoring 1) Take out any common factors. 2) Recognize if polynomial is (or isn't) a perfect square, a difference of, a difference of cubes or a sum of cubes. 3) If quadratic, try decomposition method. 4) If polynomial is higher than degree 2, find factors of the form (hx - k) or (x - k) by substituting x = k/h and or x = k into the polynomial and then using long division. 4) Continue process until polynomial is fully factored. 5) Check the factors by multiplying them together - you should get the original polynomial if the factors are correct.