Warm up Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Factoring Polynomials Digital Lesson Factoring Polynomials

Factoring these trinomials is based on reversing the FOIL process. To factor a simple trinomial of the form x2 + bx + c, express the trinomial as the product of two binomials. For example, x2 + 10x + 24 = (x + 4)(x + 6). Factoring these trinomials is based on reversing the FOIL process. Example: Factor x2 + 3x + 2. Express the trinomial as a product of two binomials with leading term x and unknown constant terms a and b. x2 + 3x + 2 = (x + a)(x + b) F O I L = x2 + bx + ax + ba Apply FOIL to multiply the binomials. = x2 + (b + a) x + ba Since ab = 2 and a + b = 3, it follows that a = 1 and b = 2. = x2 + (1 + 2) x + 1 · 2 (Product-sum method) Therefore, x2 + 3x + 2 = (x + 1)(x + 2). Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Factor x2 + bx + c

It follows that both a and b are negative. Example: Factor x2 – 8x + 15. x2 – 8x + 15 = (x + a)(x + b) = x2 + (a + b)x + ab Therefore a + b = -8 and ab = 15. It follows that both a and b are negative. Sum Negative Factors of 15 - 1, - 15 -15 -3, - 5 - 8 x2 – 8x + 15 = (x – 3)(x – 5). Check: = x2 – 8x + 15. (x – 3)(x – 5) = x2 – 5x – 3x + 15 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Factor

two positive factors of 36 13 36 Example: Factor x2 + 13x + 36. x2 + 13x + 36 = (x + a)(x + b) = x2 + (a + b) x + ab Therefore a and b are: two positive factors of 36 Sum Positive Factors of 36 whose sum is 13. 1, 36 37 2, 18 20 15 3, 12 4, 9 13 6, 6 12 = (x + 4)(x + 9) x2 + 13x + 36 Check: (x + 4)(x + 9) = x2 + 9x + 4x + 36 = x2 + 13x + 36. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Factor

Example: Factor Completely A polynomial is factored completely when it is written as a product of factors that can not be factored further. Example: Factor 4x3 – 40x2 + 100x. 4x3 – 40x2 + 100x The GCF is 4x. = 4x(x2 – 10x + 25) Use distributive property to factor out the GCF. = 4x(x – 5)(x – 5) Factor the trinomial. Check: 4x(x – 5)(x – 5) = 4x(x2 – 5x – 5x + 25) = 4x(x2 – 10x + 25) = 4x3 – 40x2 + 100x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Factor Completely

You try… Factor x2 + 7x + 12 Factor x2 – 12x + 27 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Solve equations by factoring Solve x2 + 2x = 15. x2 + 2x = 15 original equation x2 + 2x – 15 = 0 rewrite the equation so that one side equals 0 (x – 3)(x + 5) = 0 factor x – 3 = 0 or x + 5 = 0 zero product property x = 3 x = -5 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Real world problem by factoring Marion has a small art studio in her back yard. She wants to build a new studio that has three times the area of the old studio by increasing the length and width by the same amount. What are the new dimensions of the new studio? Existing studio 12 ft 10 ft x Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

x2 + 22x – 240 = 0 rewrite the equation so that one side equals 0 Current area: 12 x 10 = 120 New area: 3(120) = 360 (x + 12)(x + 10) = 360 x2 + 22x + 120 = 360 foil x2 + 22x – 240 = 0 rewrite the equation so that one side equals 0 (x + 30)(x – 8) = 0 factor x + 30 = 0 x – 8 = 0 x = -30 x = 8 Existing studio 12 ft 10 ft x length width Since we are talking about length, we know we can not have negative distance therefore our answer is x = 8 New dimensions length: 12 + 8 = 20 ft width: 10 + 8 = 18 ft Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Page 493 # 18 – 51 multiples of 3  Class work… Copyright © by Houghton Mifflin Company, Inc. All rights reserved.