Factoring (pp. 1 of 4)


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1 Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common factor (CGF). What does it mean to factor a number (or, to find its factors )? Polynomials: ( x y)(4x y) 8x xy 1xy y 8x 14xy y Consider the polynomial problem above. What are the factors? Look back at the previous activity ( Fact ors About Islands). What are the factors of x 49? What does it mean to factor a polynomial (or, to find its factors )? Factoring Polynomials Follow these steps to factor polynomials. Step One: Always look to see if the terms have a greatest common factor (GCF) (other than 1) 6x 14x 4a b 6ab 10ab GCF = GCF = ( + ) ( + + ) Step Two: After checking for the GCF, remaining polynomials can be factored by several different methods, according to the number of terms in the polynomial. A) Two Terms 1. Difference of Squares: a b ( a b)( a b) 9 64y x 0x 45 GCF = GCF = 010, TESCCC 08/01/10
2 Factoring (pp. of 4) Algebra Two Terms (continued). Difference of Cubes: a b ( a b)( a ab b ) GCF = x y. Sum of Cubes: a b ( a b)( a ab b ) 16m p GCF = B) Three Terms 1. Leading Coefficient of 1 x 4x 1 x 10x 1 GCF = GCF =. Leading Coefficient other than 1 6x 7x 5 Various Methods Guess and check GCF = Box method 010, TESCCC 08/01/10
3 Factoring (pp. of 4) Algebra Gross Product Bottoms up 010, TESCCC 08/01/10
4 Factoring (pp. 4 of 4) Algebra C) Four Terms Grouping 1xy + x + 0y + 5 6ab + a + 15b + 5 ( + ) + ( + ) ( + )( + ) Practice Problems Factor the following polynomials. 1) 14x y 4xy xy ) 4x 9 ) a ab 4) x 7 5) 64x 1 6) x y + 8y 7) x 14x 49 8) x 6x 56 9) y y 54 10) 4y 11y 11) 5a a 8 1) 6x 6x 0 1) 6x 9x 81 14) ab 9a 9b 81 15) 4xy 8x 7y 14 16) The area of a right triangle is represented by the expression 6x + 5x 4. If the height of the triangle is represented by the expression x + 4, find an expression to represent the base. 010, TESCCC 08/01/10
5 Factoring (pp. 1 of 4) KEY Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. 1,,, 4, 6, 8, 1, 4, or 1 What does it mean to factor a number (or, to find its factors )? Answers will vary. Sample: Find numbers that multiply to give you the number. C) 6 Simplify 48 using the greatest common factor (CGF). = 14 4 Polynomials: ( x y)(4x y) 8x xy 1xy y 8x 14xy y Consider the polynomial problem above. What are the factors? (x + y) and (4x + y) Look back at the previous activity ( Fact ors About Islands). What are the factors of x 49? (x + 7)(x 7) What does it mean to factor a polynomial (or, to find its factors )? Answers will vary. Sample: Find polynomials you could multiply to give you a certain answer. Factoring Polynomials Follow these steps to factor polynomials. Step One: Always look to see if the terms have a greatest common factor (GCF) (other than 1) 6x 14x 4a b 6ab 10ab GCF = x GCF = ab x (x + 7) ab (a + b + 5) Step Two: After checking for the GCF, remaining polynomials can be factored by several different methods, according to the number of terms in the polynomial. A) Two Terms 1. Difference of Squares: a b ( a b)( a b) 9 64y x 0x 45 GCF = None (other than 1) GCF = 5 (x + 8y)(x 8y) 5(4x 9) = 5(x + )(x ) 010, TESCCC 08/01/10
6 Factoring (pp. of 4) KEY Algebra Two Terms (continued). Difference of Cubes: a b ( a b)( a ab b ) x y GCF = None (other than 1) (x y)(x + xy + y). Sum of Cubes: a b ( a b)( a ab b ) 16m p Square of 1 st term Terms multiplied and take the opposite sign Square of nd term GCF = (8m + p ) = (m + p)(4m mp + p ) B) Three Terms 1. Leading Coefficient of 1 x 4x 1 x 10x 1 GCF = None (other than 1) GCF = (x + 6)(x ) (x + 5x 6) = (x + 6)(x 1) After GCF, Find factors of the last term Find factors of the last term (1), that (6), that combine to give the middle term combine to give the middle term (+4) (+5) ()(4)NO ()(4)NO ()()NO ()()NO (6)()YES (6)()NO (6)(1)YES (6)(1)NO. Leading Coefficient other than 1 6x 7x 5 Various Methods Guess and check GCF = None (other than 1) (x + 5)(x 1) Box method 010, TESCCC 08/01/10
7 Factoring (pp. of 4) KEY Algebra Gross Product (Illustrated below) Find the product of the leading coefficient and constant. 6 5 = 0 Determine two factors of this product that combine to give the middle term. Factors of 0 that combine to and  Replace the middle term with two x terms with these coefficients. 6x 10xx 5 Group as binomials. (See four terms.) 6x 10x x 5 Factor each binomial. xx 51x 5 Factor out the common factor and group the remaining terms. x5 x 1 Bottoms up (Illustrated below) Multiply leading coefficient and constant and put result in as the final term. x + 7x 0 Factor as before as you would with a leading coefficient of 1. (x + 10)(x ) Divide the 6 out of the last terms. (x + 10/6)(x /6) Simplify rational numbers. (x + 5/)(x ½) Bottoms up the remaining denominators. (x +5)(x 1) 010, TESCCC 08/01/10
8 Factoring (pp. 4 of 4) KEY Algebra C) Four Terms Grouping 1xy + x + 0y + 5 6ab + a + 15b + 5 x ( 6y + 1) + 5 (6y + 1 ) a(b + 1) + 5(b + 1) (6y + 1)(x + 5) (b + 1)(a + 5) Practice Problems Factor the following polynomials. 1) 14x y 4xy xy ) 4x 9 xy(7x + y + 1) (y )(y + ) ) a ab 4) x 7 a(a b)(a + b) (x + )(x x + 9) 5) 64x 1 6) x y + 8y (4x 1)(16x + 4x + 1) y(x + )(x x + 4) 7) x 14x 49 8) x 6x 56 (x 7)(x 7) or (x 7) (x + 7)(x 4) 9) y y 54 10) 4y 11y (y 9)(y + 6) (y + )(4y 1) 11) 5a a 8 1) 6x 6x 0 (a 4)(5a ) (6x + 5)(x ) 1) 6x 9x 81 14) ab 9a 9b 81 (x 9)(x + ) (b 9)(a + 9) 15) 4xy 8x 7y 14 (y )(4x + 7) 16) The area of a right triangle is represented by the expression 6x + 5x 4. If the height of the triangle is represented by the expression x + 4, find an expression to represent the base. 6x + 5x 4 = ½ (x + 4)b (6x + 5x 4) = (x + 4)b (x +4)(x 1) =(x + 4)b b = (x 1) or 4x 010, TESCCC 08/01/10
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