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


 Eugene Berry
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1 Factoring Factoring Polynomial Equations Ms. Laster Earlier, you learned to factor several types of quadratic expressions: General trinomial  2x 25x12 = (2x + 3)(x  4) Perfect Square Trinomial  x 2 +10x+25=(x + 5) 2 Difference of  4x 29 =(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 25x12 = (2x + 3)(x  4) Perfect Square Trinomial  x 2 +10x+25=(x + 5) 2 Difference of  4x 29 =(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 22x + 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 31 = (2x  1)(2x 2 + 2x + 1) Factor 64a 427a 64a 427a Factor common monomial 64a 427a a(64a 327) Factor common monomial 64a 427a Factor common monomial a(64a 327) Difference of cubes
3 64a 427a Factor common monomial a(64a 327) Difference of cubes a((4a) ) 64a 427a Factor common monomial a(64a 327) 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 23x 24y x 2 y 23x 24y x 2 is common to the first two terms, and 4 is common to the second two.
4 x 2 y 23x 24y x 2 (y 23)4(y 23) factored out a 4 remember, you x 2 y 23x 24y x 2 (y 23) 4(y 23) (x 24)(y 23) 1st term is a difference of x 2 y 23x 24y x 2 (y 23) 4(y 23) (x 24)(y 23) (x  2)(x + 2)(y 23) 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 416 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 24)(9x 2 + 4) (9x 2 ) Now, this is a difference of (9x 24)(9x 2 + 4) That first term is another difference of (9x 2 ) Now, this is a difference of (9x 24)(9x 2 + 4) (3x  2)(3x + 2)(9x 2 + 4) Try factoring a 2 b 28ab b 4 Common factor b 2 (a 28ab + 16b 2 )
6 b 2 (a 28ab + 16b 2 ) Perfect Square Trinomial b 2 (a 28ab + 16b 2 ) b 2 (a 24b)(a 24b) or b 2 (a 24b) 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.
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