Warm Up SUM-PRODUCT PUZZLES For each figure below, find the pair of numbers whose product is the top number in the X and whose sum is the bottom number in the X. 14 14 -33 2 7 -2 -7 -3 11 9 -9 8 -4 56 6 2 -2 -8 -7 -2 -3 -15 -5

Math 8H 8-3 Factoring Trinomials x2 + bx + c Algebra 1 Glencoe McGraw-Hill JoAnn Evans

The Sum-Product Puzzles you did in the warm-up are a useful tool when factoring quadratic trinomials. Today we’ll factor quadratic trinomials that have a leading coefficient of 1. They’ll look like this: ax2 + bx + c. The number that replaces the variable “a” will always be 1 in this lesson.

Put b in the bottom. It will be the sum of the two side numbers. ax2 + bx + c To use this tool, draw an X next to each problem. Multiply a  c and put the product in the top of the figure. This number will be the product of the two side numbers. Put b in the bottom. It will be the sum of the two side numbers. a●c b

Factor: x2 – 8x + 12 ax2 + bx + c 12 -2 -6 -8 a = 1 b = -8 c = 12 In this trinomial, what are the values of the variables a, b, and c? -8 b a = 1 b = -8 c = 12 To fill in the sides of the X figure you need to find two numbers that have a product of 12 and a sum of -8. What would those numbers be?

a●c Place the numbers from the sides of the X figure into your factored answer this way: (x ) (x ) 12 -2 -6 -8 – 2 – 6 b Side # Side # Check your answer by FOILing in your head. Look carefully at the sum of the Inner ∙ Inner + Outer ∙ Outer to check that their sum equals the middle term of the trinomial you just factored. (x – 2) (x – 6) = x2 – 6x – 2x + 12 = x2 – 8x + 12

(x ) (x ) – 7 + 3 Factor: x2 – 4x - 21 ax2 + bx + c -21 -7 3 In this trinomial, what are the values of the variables a, b, and c? -4 b a = 1 b = -4 c = -21 Find two numbers that have a product of -21 and a sum of -4. (x ) (x ) – 7 + 3 Side # Side # Check: (x - 7) (x + 3) = x2 + 3x – 7x – 21 = x2 - 4x - 21

(x ) (x ) + 2 + 13 Factor: x2 + 15x + 26 ax2 + bx + c 26 2 13 In this trinomial, what are the values of the variables a, b, and c? 15 b a = 1 b = 15 c = 26 Find two numbers that have a product of 26 and a sum of 15. (x ) (x ) + 2 + 13 Check: (x + 2) (x + 13) = x2 + 13x + 2x + 26 = x2 + 15x + 26

(x ) (x ) + 2 - 3 Factor: x2 - x - 6 ax2 + bx + c -6 2 -3 In this trinomial, what are the values of the variables a, b, and c? -1 b a = 1 b = -1 c = -6 Find two numbers that have a product of -6 and a sum of -1. (x ) (x ) + 2 - 3 Check: (x + 2) (x - 3) = x2 - 3x + 2x - 6 = x2 - x - 6

Find two monomials that have a product of 10y2 and a sum of 7y. Factor: x2 + 7xy + 10y2 a●c 10y2 Think of it as: x2 + 7yx + 10y2 2y 5y ax2 + bx + c 7y b a = 1 b = 7y c = 10y2 Find two monomials that have a product of 10y2 and a sum of 7y. (x ) (x ) + 2y + 5y Check: (x + 2y) (x + 5y) = x2 + 5xy + 2xy + 10y2 = x2 + 7xy + 10y2

Time for 18 Practice Problems!