Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 4.2 Factoring Trinomials of the Form x 2 + bx + c

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 22 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Factoring Trinomials of the Form x 2 + bx + c Recall by using the FOIL method that F O I L (x + 2)(x + 4) = x 2 + 4x + 2x + 8 = x 2 + 6x + 8 To factor x 2 + bx + c into (x + one #)(x + another #), note that b is the sum of the two numbers and c is the product of the two numbers. So we’ll be looking for 2 numbers whose product is c and whose sum is b. Note: there are fewer choices for the product, so that’s why we start there first.

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 33 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. To Factor a Trinomial of the Form x 2 + bx + c Factoring Trinomials of the Form x 2 + bx + c The product of these numbers is c. The sum of these numbers is b. x 2 + bx + c = (x + )(x + )

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Factor the polynomial x x Since our two numbers must have a product of 30 and a sum of 13, the two numbers must both be positive. Positive factors of 30Sum of Factors 1, , , 1013 Note, there are other factors, but once we find a pair that works, we do not have to continue searching. So x x + 30 = (x + 3)(x + 10). Factoring Polynomials Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 55 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Factor the polynomial x 2 – 11x Since our two numbers must have a product of 24 and a sum of -11, the two numbers must both be negative. Negative factors of 24Sum of Factors – 1, – 24 – 25 – 2, – 12 – 14 – 3, – 8 – 11 So x 2 – 11x + 24 = (x – 3)(x – 8). Factoring Polynomials Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 66 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Factor the polynomial x 2 – 2x – 35. Since our two numbers must have a product of – 35 and a sum of – 2, the two numbers will have to have different signs. Factors of – 35Sum of Factors – 1, , – 35 – 34 – 5, 7 2 5, – 7 – 2 So x 2 – 2x – 35 = (x + 5)(x – 7). Factoring Polynomials Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 77 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Factor: x 2 – 6x + 10 We look for two numbers whose product is 10 and whose sum is – 6. The two numbers will have to both be negative. Negative factors of 10Sum of Factors – 1, – 10 – 11 – 2, – 5 – 7 Since there is not a factor pair whose sum is – 6, x 2 – 6x +10 is not factorable and we call it a prime polynomial. Prime Polynomials Example

Martin-Gay, Prealgebra & Introductory Algebra, 3ed 88 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. You should always check your factoring results by multiplying the factored polynomial to verify that it is equal to the original polynomial. Many times you can detect computational errors or errors in the signs of your numbers by checking your results. Check Your Result!