Math 100 Greatest Common Factor and Factoring by Grouping (Review) Factoring Definition: A factor is a number, variable, monomial, or polynomial which is multiplied by another number, variable, monomial, or polynomial to obtain a product. 1. List all the possible factors of the following numbers: a. 1 b. c. 19 d. In the above, the number 19 is an example of a number because its only positive factors are one and itself. (Review) Greatest Common Factor Definition: The greatest common factor of two or more numbers is the largest number that divides (goes into) the given numbers with a remainder of zero.. Find the GCF (greatest common factor) of the following numbers: a. and b. 1 and 0 c. 1,, and d. 8 and 9 1
Math 100 Greatest Common Factor of Polynomials In order to find the GCF of two or more monomials, I. Find the GCF of the coefficients; II. Find the GCF of the variables; III. Rewrite the GCF as a product of the GCF of the coefficients times the GCF of the variables. 1. Find the GCF of x and x 7. Step I: The only coefficients are 1 s, so this is the GCF of the coefficients. Step II: Rewrite the two monomials as products of xs without using exponents: x = x x x x x 7 = x x x x x x x Since each monomial has xs in it, the GCF of the variables is x. Step III: The GCF of x and x 7 is x.. Find the GCF of xy and x y. Step I: The only coefficients are 1 s, so this is the GCF of the coefficients. Step II: Rewrite the two monomials as products of xs and ys without using exponents: xy = x y y x y = x x x x x y y y y Since each monomial has 1 x and ys in it, the GCF of the variables is xy. Step III: The GCF of xy and x y is xy.. Find the GCF of x y and 9x 7 y. Step I: The GCF of the coefficients and 9 is. Step II: Rewrite the two monomials as products of xs and ys without using exponents: x y = x x x x y x 7 y = x x x x x x x y y y y Since each monomial has xs and 1 y in it, the GCF of the variables is x y. Step III: The GCF of x y and 9x 7 y is x y. 1. Find the GCF of 18 x and 7x.. Find the GCF of x and 9x.. Find the GCF of 10x and 0 x.
Math 100. Find the GCF of 1xy and 1x y. Find the GCF of 8x 9 and x y.. Find the GCF of a b c and 0a bc. 7. Find the GCF of 8 x, 1x and x. 8. Find the GCF of 9x, 1x and 7 x. 9. Find the GCF of 8xy, x y and 10x y. 10. Find the GCF of a b, 1a b and 1a b.
Math 100 Factoring To factor a polynomial, try to find the GCF of the monomials in the polynomial. Then this GCF should be factored out of the polynomial by undistributing the GCF out of all the monomials in the polynomial. Note that if the leading coefficient is negative, then the GCF should also be negative. 1. Factor 7x + 1y. Step I: Find the GCF of the coefficients. GCF(7, 1) = 7. Step II: Find the GCF of the variable parts. There is none, so nothing changes. Step III: Divide all the monomials by the GCFs and rewrite with the GCFs out front: 7x 1y 7x + 1y = 7 + = 7(x + y) 7 7. Factor 8x + x. Step I: Find the GCF of the coefficients. GCF(8, ) =. Step II: Find the GCF of the variable parts. GCF(x, x ) = x. Step III: Divide all the monomials by the GCFs and rewrite with the GCFs out front: 8x + x 8x x = x + = x (x + ) x x 1. Factor x + 7x. Factor 8x + x. Factor 1x x. Factor 1x + 8x. Factor 1x y + x y
Math 100. Factor ab + a. 7. Factor 8a bc 1ab c. 8. Factor -8a b 7 a b. 9. Factor 1x y + xy 1x y 10. Factor 1x + x 8x 11. Factor 10a b + ab c 8a b c 1. Factor y ( x 1) + ( x 1) 1. Factor x ( y + ) z( y + ) 1. Factor x ( y ) + x ( y )
Math 100 Factoring by Grouping If a polynomial contains four or more terms, it may be helpful to put the terms into groups of two and factor out a common factor from each of these groups. This is called grouping. 1. Factor x y + x + xy + 18y Step I: Group the terms so that each group shares a common factor: x y + x + xy + 18y = (x y + x) + (xy + 18y) Step II: Factor out the common terms from each group: (x y + x) = x(xy + ) (xy + 18y) = y(xy + ) Step III: Rewrite the polynomial as the sum of the factored groups: x y + x + xy + 18y = x(xy + ) + y(xy + ) Step IV: Factor the resulting polynomial from Step III: x(xy + ) + y(xy + ) = (xy + )(x + y). Factor x + x + x + Step I: Group the terms so that each group shares a common factor: x + x + x + = (x + x ) + (x + ) Step II: Factor out the common terms from each group: (x + x ) = x (x + ) (x + ) = 1(x + ) (Note: There s no common term, so the GCD is 1.) Step III: Rewrite the polynomial as the sum of the factored groups from Step II. x + x + x + = x (x + ) + 1(x + ) Step IV: Factor out the resulting polynomial from Step III: x (x + ) + 1(x + ) = (x + 1)(x + ). Factor x + x x. Step I: Group the terms so that each group shares a common factor: x + x x = (x + x ) + (-x ) Step II: Factor out the common terms from each group: (x + x ) = x (x + 1) (-x ) = -(x + 1) Step III: Rewrite the polynomial as the sum of the factored groups from Step II. x + x x = x (x + 1) + (-(x + 1)) = x (x + 1) (x + 1) Step IV: Factor out the resulting polynomial from Step III: x (x + 1) (x + 1) = (x )(x + 1)
Math 100 1. Factor a + 8a + a b + b.. Factor a x + a y 1x y.. Factor ax + xb + 10ay + 10yb.. Factor x + 9xy + 10y + 1y. 7