Factoring Factoring Polynomial Equations Ms. Laster Algebra II General trinomial - 2x2-5x-12 = (2x + 3)(x - 4) Perfect Square Trinomial - x2+10x+25=(x + 5)2 Difference of squares - 4x2 - 9 =(2x + 3)(2x - 3) Common monomial factor - 6x2+15x=3x(2x+5) Algebra II Factoring • Earlier, you learned to factor several types of quadratic expressions: • Earlier, you learned to factor several types of quadratic expressions: General trinomial - 2x2-5x-12 = (2x + 3)(x - 4) Perfect Square Trinomial - x2+10x+25=(x + 5)2 Difference of squares - 4x2 - 9 =(2x + 3)(2x - 3) Common monomial factor - 6x2+15x=3x(2x+5) Special Factoring Patterns • Two common patterns are the difference and sum of two cubes. • Now, we’ll look at some other types of polynomials. Algebra II Special Factoring Patterns • Two common patterns are the difference and sum of two cubes. • Sum of two cubes a3 + b3 = (a + b)(a2 - ab + b2) Algebra II Algebra II Special Factoring Patterns • Two common patterns are the difference and sum of two cubes. • Sum of two cubes a3 + b3 = (a + b)(a2 - ab + b2) x3 + 8 = (x + 2)(x2 - 2x + 4) Algebra II 1 Special Factoring Patterns • Two common patterns are the difference and sum of two cubes. • Difference of two cubes a3 - b3 = (a - b)(a2 + ab + b2) Algebra II Special Factoring Patterns • Factor 64a4 - 27a Algebra II Special Factoring Patterns • 64a4 - 27a Factor common monomial • a(64a3 - 27) Algebra II Special Factoring Patterns • Two common patterns are the difference and sum of two cubes. • Difference of two cubes a3 - b3 = (a - b)(a2 + ab + b2) 8x3 - 1 = (2x - 1)(2x2 + 2x + 1) Algebra II Special Factoring Patterns • 64a4 - 27a Factor common monomial Algebra II Special Factoring Patterns • 64a4 - 27a Factor common monomial • a(64a3 - 27) Difference of cubes Algebra II 2 Special Factoring Patterns • 64a4 - 27a Factor common monomial • a(64a3 - 27) Difference of cubes • a((4a)3 - 33) Algebra II Special Factoring Patterns • • • • 64a4 - 27a Factor common monomial a(64a3 - 27) Difference of cubes a((4a)3 - 33) a(4a - 3)(16a2 +12a + 9) Algebra II Factoring by Grouping Factoring by Grouping • 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) Algebra II Factoring by Grouping • Factor x2y2 - 3x2 - 4y2 + 12 Algebra II Factoring by Grouping • x2y2 - 3x2 - 4y2 + 12 x2 is common to the first two terms, and 4 is common to the second two. Algebra II Algebra II 3 Factoring by Grouping • x2y2 - 3x2 - 4y2 + 12 • x2(y2 - 3) - 4(y2 - 3) remember, you factored out a -4 Factoring by Grouping • x2y2 - 3x2 - 4y2 + 12 • x2(y2 - 3) - 4(y2 - 3) • (x2 - 4)(y2 - 3) 1st term is a difference of squares Algebra II Factoring by Grouping • • • • x2y2 - 3x2 - 4y2 + 12 x2(y2 - 3) - 4(y2 - 3) (x2 - 4)(y2 - 3) (x - 2)(x + 2)(y2 - 3) Algebra II Algebra II Factoring Polynomials in Quadratic Form • Sometimes an expression will be in quadratic form, but not obviously. Any expression in the form au2 + bu + c, where u is some expression of x, is quadratic. Algebra II Factoring Polynomials in Quadratic Form Factoring Polynomials in Quadratic Form • Sometimes an expression will be in quadratic form, but not obviously. Any expression in the form au2 + bu + c, where u is some expression of x, is quadratic. • 81x4 - 16 This is not obviously quadratic, since 81x4 doesn’t look like a perfect square, but let’s look again. • 81x4 - 16 • (9x2)2 - 42 Now, this is a difference of Algebra II squares Algebra II 4 Factoring Polynomials in Quadratic Form Factoring Polynomials in Quadratic Form • 81x4 - 16 • (9x2)2 - 42 Now, this is a difference of • 81x4 - 16 • (9x2)2 - 42 Now, this is a difference of squares • (9x2 - 4)(9x2 + squares 4) • (9x2 - 4)(9x2 + 4) That first term is another difference of squares Algebra II Algebra II Factoring Polynomials in Quadratic Form Factoring Polynomials in Quadratic Form • 81x4 - 16 • (9x2)2 - 42 Now, this is a difference of • Try factoring a2b2 - 8ab3 + 16b4 squares • (9x2 - 4)(9x2 + 4) • (3x - 2)(3x + 2)(9x2 + 4) Algebra II Algebra II Factoring Polynomials in Quadratic Form Factoring Polynomials in Quadratic Form • a2b2 - 8ab3 + 16b4 Common factor • a2b2 - 8ab3 + 16b4 • b2(a2 - 8ab + 16b2) Algebra II Algebra II 5 Factoring Polynomials in Quadratic Form Factoring Polynomials in Quadratic Form • a2b2 - 8ab3 + 16b4 • b2(a2 - 8ab + 16b2) Perfect Square • a2b2 - 8ab3 + 16b4 • b2(a2 - 8ab + 16b2) • b2(a2 - 4b)(a2 - 4b) or Trinomial Algebra II b2(a2 - 4b)2 Algebra II 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 squares, 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. Algebra II 6

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