The greatest common factor, or GCF, is the largest factor that two or more terms share.

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1 Unit, Lesson Factoring Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product You can factor certain expressions by writing them as the product of factors The Zero Product Property states that if the product of two factors is 0, then at least one of the factors is 0 After setting a quadratic equation equal to 0, you can sometimes factor the quadratic expression and solve the equation by setting each factor equal to 0 The greatest common factor, or GCF, is the largest factor that two or more terms share You should always check to see if the terms of an expression have a greatest common factor before attempting to factor further The value of a for a quadratic expression in the form ax + bx + c is called the leading coefficient, or lead coefficient, because it is the coefficient of the term with the highest power To factor a trinomial with a leading coefficient of 1 in the form x + bx + c, find two numbers d and e that have a product of c and a sum of b The factored form of the expression will be (x + d)(x + e) When finding d and e, be careful with the signs The table that follows shows that the signs of d and e will be based on the signs of b and c Signs of b, c, d, and e b c d e Opposite signs; the number with the larger absolute value is positive Opposite signs; the number with the larger absolute value is negative You may be able to factor expressions with lead coefficients other than 1 in your head or by using guess-and-check Expressions with leading coefficients other than 1 in the form factored by grouping ax b c can sometimes be If you struggle to factor expressions in your head or by using guess-and-check, factoring by grouping is a more structured alternative Factor by Grouping 1 Begin by finding two numbers d and e whose product is ac and whose sum is b Rewrite the expression by replacing bx with dx + ex: ax d e c Factor the greatest common factor from ax dx 4 Factor the greatest common factor from e c 5 Factor the greatest common factor from the resulting expression MUnitLesson 1 6/8/018

2 Unit, Lesson Factoring (continued) A quadratic expression in the form ax ) b ( is called a difference of squares The difference of squares ax ) b ( can be written in factored form as ( a b)(a b) Some expressions cannot be factored These expressions are said to be prime Although the difference of squares is factorable, the sum of squares is prime For example, ( ) ( 5)( 5), but ( ) is not factorable Common Errors/Misconceptions forgetting to consider the signs of a, b, and c treating a leading coefficient other than 1 as if it were a 1 multiplying the terms of an expression given in factored form to solve an equation solving an expression that is not part of an equation confusing the difference of squares with the sum of squares Example 1 Factor x The leading coefficient is 1, so begin by finding two numbers whose product is ac 15 and whose sum is b 8 The numbers are and 5 because ( )( 5) 15 and ( 5) 8 Find the factors Use and 5 to find the factors The factors will be ( ) and ( 5) Write the expression as the product of its factors The factors are ( ) and ( 5), so the product of the factors is ( )( 5) The product should be the original expression ( )( 5) 815 Example Solve by factoring 1 Rewrite the equation so that all terms are on one side of the equation Original equation 9x 8 56 Add x to both sides 9x 64 0 Subtract 56 from both sides MUnitLesson 6/8/018

3 Unit, Lesson Factoring (continued) Factor the difference of squares The expression on the left side can be rewritten in the form ( ) 8 You can use this form to rewrite the expression as the difference of squares to factor the expression ( 8)( 8) 0 Use the Zero Product Property to solve Example The expression will equal 0 only when one of the factors is equal to 0 Set each factor equal to 0 and solve 8 0 or So, when Solve x 8 0 by factoring 8 or 8 1 Rewrite the equation so that all terms are on one side of the equation x x 8 0 Original equation Subtract 0 from both sides Find the factors The leading coefficient of the expression is 1, so begin by finding two numbers whose product is ac 0 and whose sum is b 8 The numbers are and 10 because ( )(10 ) 0 and 10 8 Therefore, the factors are ( ) and ( 10 ) Write the expression as the product of its factors ( )( 10 ) 0 4 Use the Zero Product Property to solve the equation The expression will equal 0 only when one of the factors is equal to 0 Set each factor equal to 0 and solve 0 or 10 0 or 10 So, x 8 0 when or 10 MUnitLesson 6/8/018

4 Unit, Lesson Factoring (continued) Example 4 Factor 10 by grouping 1 Find two numbers whose product is ac and whose sum is b The expression is in the form ax b c a, b 1, and c 10 ac ()( 10 ) 0 You need to find two numbers whose product is 0 and whose sum is 1 The numbers are 6 and 5 because ( 6)(5) 0 and Replace bx in the original equation The numbers found above have a sum of b, so you can replace bx with 10 Original equation ( 6x 5x ) 10 Rewritten equation 6x 5x Factor the greatest common factor of the two left-hand terms Do the same with the two right-hand terms The greatest common factor of the left-hand terms is ( ) 510 The greatest common factor of the right-hand terms is 5 ( ) 5( ) 4 Factor the greatest common factor of the expression Example 5 The greatest common factor of the expression is ( ) 10 ( )( 5) You have just factored by grouping You can check your work by multiplying the factors The product should be the original expression Solve 7x Factor out the greatest common factor of the expression The greatest common factor is 7 7x ( 910 ) 0 Find two numbers whose product is 10 and whose sum is 9 The numbers are 1 and 10 because ( 1)(10 ) 10 and Use these numbers to find the factors The factors are ( 1) and ( 10 ) 4 Write the expression as the product of its factors 7( 1)( 10 ) 0 MUnitLesson 4 6/8/018

5 Unit, Lesson Factoring (continued) Example 5 (continued) 5 Use the Zero Product Property to solve The expression will equal 0 only when one of the factors is equal to 0 The factor of 7 will never equal 0 Set each of the remaining factors equal to 0 and solve 1 0 or So, 7x when 1 or when 10 Example 6 Solve 8x Rewrite the equation so that all the terms are on one side of the equation 8x 18 5 Original equation 8x Subtract 5 from both sides Factor the equation The leading coefficient is not 1, so factor by grouping Begin by finding two numbers whose product is ac and whose sum is b a 8, b 18, and c 5 ac (8)( 5) 40 Find two numbers whose product is 40 and whose sum is 18 The numbers are 0 and because ( 0 )( ) 40 and 0 18 Replace bx in the original expression The numbers you found have a sum of b, so you can replace bx with 8x ( 0 x ) 5 0 (8 0x ) ( 5) 0 0x x Factor out the greatest common factor of the left-hand terms Do the same with the righthand terms The greatest common factor of the left-hand terms is 4x 4x( 5) ( 5) 0 The greatest common factor of the right-hand terms is 1 4x( 5) 1( 5) 0 Factor out the greatest common factor of the expression, ( 5) ( 5)( 41) 0 MUnitLesson 5 6/8/018

6 Unit, Lesson Factoring (continued) Example 6 (continued) Set each factor equal to 0 and solve x 5 0 or 4x 1 0 x 5 4x So, 8x 18 5 when 5 or when 1 4 MUnitLesson 6 6/8/018

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