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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.

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### expression is written horizontally. The Last terms ((2)( 4)) because they are the last terms of the two polynomials. This is called the FOIL method. A polynomial of degree n (in one variable, with real coefficients) is an expression of the form: a n x n + a n 1 x n 1 + a n 2 x n 2 + + a 2 x 2 + a 1 x + a 0 where a n, a n 1, a n 2, a 2, a 1, a 0 are

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### 1.5. Factorisation. Introduction. Prerequisites. Learning Outcomes. Learning Style Factorisation 1.5 Introduction In Block 4 we showed the way in which brackets were removed from algebraic expressions. Factorisation, which can be considered as the reverse of this process, is dealt with Polynomials and Quadratics Want to be an environmental scientist? Better be ready to get your hands dirty!.1 Controlling the Population Adding and Subtracting Polynomials............703.2 They re Multiplying

### Mathematics Placement Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry. 9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : Module HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,
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