a*(variable) 2 + b*(variable) + c

Transcription

1 CH. 8. Factoring polynomials of the form: a*(variable) + b*(variable) + c Factor: 6x + 11x + 4 STEP 1: Is there a GCF of all terms? NO STEP : How many terms are there? Is it of degree? YES * Is it in the form a*(variable) + b*(variable) + c? YES In this example a = 6, b=11, c = 4 The trick for these trinomials is to multiply a (the coefficient of x ) to c, the constant term, 4. ac = 6*4 = 4. Next, find a pair of factors of that number that add up to the middle term s coefficient. Factors of 4 Sum of those Factors 1, = 5, = 14, = 11 4, = 10 Since + 8 = 11, so let s use those factors and rewrite the middle term, 11x, as x + 11x. 6x + 11x + 4 = 6x + x + 8x + 4 Now we have 4 terms, let s factor by grouping. = x(x + 1) + 4(x + 1) common factor = (x + 1)(x + 4)

2 BOX METHOD 6x + 11x + 4 As before, find a pair of factors of 4 (since 6* 4 = 4) that add up to the middle term s coefficient, 11. We already figured out that * 8 = 4 and x + 8x = 11, so our factors to use are and 8. First, make a box with the first term, 6x in the upper left corner and then last term term, 4 in the lower left corner. GCF s: x 1 x 6x x 4 8x 4 Then put in the factors multiplied by x in the other boxes (it doesn t matter which ones). That is, we will put x and 8x in the other boxes. We then proceed to find the GCF of each row and each column of the box. If there is no common factor, just use 1. Now use these GCF s for your factorization: (x + 4)(x + 1)!!!

3 Example 5 Factor: 8y 10y - STEP 1: Is there a GCF of all terms? NO STEP : How many terms are there? Is it of degree? YES * Is it in the form a*(variable) + b*(variable) + c? YES In this example, a = 8, b = -10, c = - ac = 8*- = -4 b = -10 What pair of factors of -4 will add up to -10? In the previous example we only had to look at each pair once since the last term ac was positive and the middle term, b, was also positive. Now in this example, a is positive (8) and c is negative (-) so ac= -4, which is negative. The middle term s coefficient, b, is Factors of 4 Sum of those Factors -1, = 1, = -, = -10 -, = 10 -, = 5, = -5 4, = - -4, = So we will split the middle term, -10y in to y + -1y

4 What was that polynomial again? 8y 10y - =8y + y + -1y - =y(4y + 1) + -(4y+ 1) =(4y + 1)(y ) BOX METHOD: GCF s: 4y 1 y - 8y -1y y - Since both terms in the left column have a negative coefficient, factor out a negative number. FACTORIZATION: (4y + 1)(y - )

5 Example 7 Factor: 4x y 76xy + 40y STEP 1: Is there a GCF of all terms? YES. GCF=4y Factor out 4y from the polynomial. 4y(6x - 19x + 10) STEP : How many terms are there? Is it of degree? YES * Is it in the form a*(variable) + b*(variable) + c? YES In this example, a = 6, b = -19, c = 10 ac = 6*10 = 60 b = -19 Since b is negative and ac is positive, both factors of ac must be negative in order for the product to be positive and the sum to be negative. Factors of 4 Sum of those Factors -1, = -61 -, = - -, = - -4, = -19-5, = -17-6, = -16 So we will split the middle term, -19x in to -4x + -15x

6 Let s do the grouping method this time: 4y(6x - 19x + 10) Let s just work inside the parentheses for now, but don t forget that 4y at the end! Inside the parentheses: 6x + -4x + -15x + 10 = x(x - ) + -5(x ) = (x )(x 5) COMPLETE FACTORIZATION: 4y(x )(x 5)

7 8.4 SPECIAL FACTORING Remember these? (a + b) = a + ab + b (a b) = a ab + b When you see a trinomial that starts and ends with a perfect square, it s possible the factorization could be a square of a binomial. Example: Factor: 4x 0x + 5 STEP 1: Is there a GCF of all the terms? NO STEP : How many terms are there? Is it of degree? YES Are the first and last terms perfect squares? YES 4x can be rewritten as (x), so it is a perfect square. 5 can be rewritten as 5, so it is a perfect square. The bases of those squares are x and 5. The middle term of a trinomial can be factored into the square of a binomial is *base of the first term * base of the second term. The middle term is -0x = -(4x)(5). So this trinomial is the square of the DIFFERENCE OF THE BASES (since the middle term is negative). 4x 0x + 5 = (x 5) Check: (x 5) =(x 5)(x 5)= = (x)(x)+ (-5)(x) + -5(x) + (-5)(-5) = 4x + -10x + -10x + 5 = 4x 0x + 5

8 The Difference of Squares Recall this one: (a b)(a + b) = a ab + ab b = a b So going the other way, a b can be factored into (a - b)(a + b) Example: Factor 5x y STEP 1: Is there a GCF of all terms? NO STEP : How many terms are there? Check if this is a difference of two squares. 5x = (5x) y = (y) SO. 5x y = (5x y)(5x + y) CHECK: (5x y)(5x + y) = 5x - 5xy + 5xy + (-y)(y) = 5x y

9 Ch. 8.5 Solving Quadratic Equations A quadratic equation is an equation in the form: ax + bx + c = 0 Notice this is a trinomial that is set equal to 0. If this trinomial can be factored, we can use the Principle of Zero Products to solve this equation. Principle of Zero Products If the product of two factors is zero, then at least one of the factors must be zero. That is, If a*b = 0, then a=0 or b=0. Example 1: Solve x + x = 6 This does not look at first like a quadratic equation, but if we subtract 6 from both sides, we will have a zero on the right side. x + x 6 = 0 Now factor the polynomial. Is there a GCF of all terms? NO How many terms are there? Is it degree? YES Is it the form ax + bx + c? YES. a=, b=1, c=-6 ac =*-6 = -1 b =1 Use factors - and 4 since -+4 = 1 x -x+4x 6 = 0 x(x-) + (x-) = 0 (x-)(x+) = 0 The factors are x and x+. If x = 0 Then x = x = / If x + = 0 x = - The possible solutions are x = / and x= - This solution set can be written in braces, not ()., CHECK : x x 9 (4) ? 6? ( 6 ) 6

10 Example : Solve: (x )(x-0) = -10 We can t use the Zero Product Property yet because this product = -10, not 0. We must expand it, get everything on the left hand side and zero on the right hand side, then re-factor it. Example : The sum of the squares of two consecutive positive odd integers is equal to 10. Find the two integer. What are we being asked tofind? Two consective positive integers. Let n = first integer Let n + = second integer. (If n=, n+ = 5, if n=5, n+ = 7, etc ) Information: The sum of the squares of the two integers is 10. n + (n+) = 10 n + n + 4n + 4 = 10 Simplify.. n + 4n + 4 = 10 Get everything on the left hand side. n + 4n - 16 = 0 Is there a GCF of all terms? YES: (n + n - 6) = 0 How many terms are inside the parentheses? What degree? What factors of -6 can be added to get the middle term,? ( n )( n )= 0