Introduction Recall that a factor is one of two or more numbers or expressions that when multiplied produce a given product. We 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, we can sometimes factor the quadratic expression and solve the equation by setting each factor equal to 0. 1 5.2.2: Factoring Key Concepts • 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 ax2 + bx + c is called the leading coefficient, or lead coefficient, because it is the coefficient of the term with the highest power. 2 5.2.2: Factoring Key Concepts, continued • To factor a trinomial with a leading coefficient of 1 in the form x2 + 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 what the signs of d and e will be based on the signs of b and c. 3 5.2.2: Factoring Key Concepts, continued 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. 4 5.2.2: Factoring Key Concepts, continued • You may be able to factor expressions with lead coefficients other than 1 in your head or by using guess-and-check. • Expressions with lead coefficients other than 1 in the form ax2 + bx + c can sometimes be factored by grouping. • If you struggle to factor expressions in your head or by using guess-and-check, factoring by grouping is a more structured alternative. 5 5.2.2: Factoring Key Concepts, continued Factoring by Grouping 1. Begin by finding two numbers d and e whose product is ac and whose sum is b. 2. Rewrite the expression by replacing bx with dx + ex: ax2 + dx + ex + c. 3. Factor the greatest common factor from ax2 + dx. 4. Factor the greatest common factor from ex + c. 5. Factor the greatest common factor from the resulting expression. 6 5.2.2: Factoring Key Concepts, continued • A quadratic expression in the form (ax)2 – b2 is called a difference of squares. • The difference of squares (ax)2 – b2 can be written in factored form as (ax + b)(ax – 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, (3x)2 – 52 = 9x2 – 25 = (3x + 5)(3x – 5), but (3x)2 + 52 = 9x2 + 30x + 25 and is not factorable. 7 5.2.2: Factoring Common Errors/Misconceptions • forgetting to consider the signs of a, b, and c • treating a lead 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 8 5.2.2: Factoring Guided Practice Example 2 Solve 8x2 – 8 = –x2 + 56 by factoring. 9 5.2.2: Factoring Guided Practice: Example 2, continued 1. Rewrite the equation so that all terms are on one side. 8x2 – 8 = –x2 + 56 Original equation 9x2 – 8 = 56 Add x2 to both sides. 9x2 – 64 = 0 Subtract 56 from both sides. 10 5.2.2: Factoring Guided Practice: Example 2, continued 2. Factor the difference of squares. The expression on the left side can be rewritten in the form (3x)2 – 82. We can use this form to rewrite the expression as the difference of squares to factor the expression. (3x + 8)(3x – 8) = 0 11 5.2.2: Factoring Guided Practice: Example 2, continued 3. Use the Zero Product Property to solve. The expression will equal 0 only when one of the factors is equal to 0. Set each factor equal to 0 and solve. 3x + 8 = 0 3x = -8 8 x=3 3x - 8 = 0 3x = 8 8 x= 3 8 8 8x2 – 8 = –x2 + 56 when x = - or . 3 3 5.2.2: Factoring ✔ 12 Guided Practice: Example 2, continued 13 5.2.2: Factoring Guided Practice Example 3 Solve x2 + 8x = 20 by factoring. 14 5.2.2: Factoring Guided Practice: Example 3, continued 1. Rewrite the equation so that all terms are on one side of the equation. x2 + 8x = 20 Original equation x2 + 8x – 20 = 0 Subtract 20 from both sides. 15 5.2.2: Factoring Guided Practice: Example 3, continued 2. Find the factors. The lead coefficient of the expression is 1, so begin by finding two numbers whose product is –20 and whose sum is 8. The numbers are –2 and 10 because (–2)(10) = –20 and –2 + 10 = 8. Therefore, the factors are (x – 2) and (x + 10). 16 5.2.2: Factoring Guided Practice: Example 3, continued 3. Write the expression as the product of its factors. (x – 2)(x + 10) = 0 17 5.2.2: Factoring Guided Practice: Example 3, continued 4. Use the Zero Product Property to solve. The expression will equal 0 only when one of the factors is equal to 0. Set each factor equal to 0 and solve. x–2=0 x + 10 = 0 x=2 x = –10 x2 + 8x = 20 when x = 2 or x = –10. ✔ 18 5.2.2: Factoring Guided Practice: Example 3, continued 19 5.2.2: Factoring