Download presentation

1
**Factoring Polynomials**

Digital Lesson Factoring Polynomials

2
**Greatest Common Factor**

The simplest method of factoring a polynomial is to factor out the greatest common factor (GCF) of each term. Example: Factor 18x3 + 60x. 18x3 = 2 · 3 · 3 · x · x · x Factor each term. = (2 · 3 · x) · 3 · x · x 60x = 2 · 2 · 3 · 5 · x = (2 · 3 · x) · 2 · 5 GCF = 6x Find the GCF. 18x3 + 60x = 6x (3x2) + 6x (10) Apply the distributive law to factor the polynomial. = 6x (3x2 + 10) Check the answer by multiplication. 6x (3x2 + 10) = 6x (3x2) + 6x (10) = 18x3 + 60x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Greatest Common Factor

3
**A common binomial factor can be factored out of certain expressions.**

Example: Factor 4x2 – 12x + 20. GCF = 4. = 4(x2 – 3x + 5) Check the answer. 4(x2 – 3x + 5) = 4x2 – 12x + 20 A common binomial factor can be factored out of certain expressions. Example: Factor the expression 5(x + 1) – y(x + 1). 5(x + 1) – y(x + 1) = (x + 1) (5 – y) (x + 1) (5 – y) = 5(x + 1) – y(x + 1) Check. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Factor

4
**D.O.T.S. A difference of two squares can be factored using the formula**

a2 – b2 = (a + b)(a – b). Example: Factor x2 – 9y2. x2 – 9y2 = (x)2 – (3y)2 Write terms as perfect squares. = (x + 3y)(x – 3y) Use the formula. The same method can be used to factor any expression which can be written as a difference of squares. Example: Factor (x + 1)2 – 25y 4. (x + 1)2 – 25y 4 = (x + 1)2 – (5y2)2 = [(x + 1) + (5y2)][(x + 1) – (5y2)] = (x y2)(x + 1 – 5y2) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Difference of Squares

5
**Examples: 1. Factor 2xy + 3y – 4x – 6.**

Some polynomials can be factored by grouping terms to produce a common binomial factor. Examples: 1. Factor 2xy + 3y – 4x – 6. Notice the sign! 2xy + 3y – 4x – 6 = (2xy + 3y) – (4x + 6) Group terms. = y (2x + 3) – 2(2x + 3) Factor each pair of terms. = (2x + 3) ( y – 2) Factor out the common binomial. 2. Factor 2a2 + 3bc – 2ab – 3ac. 2a2 + 3bc – 2ab – 3ac = 2a2 – 2ab + 3bc – 3ac Rearrange terms. = (2a2 – 2ab) + (3bc – 3ac) Group terms. = 2a(a – b) + 3c(b – a) Factor. = 2a(a – b) – 3c(a – b) b – a = – (a – b). = (a – b) (2a – 3c) Factor. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Examples: Factor

6
**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

7
**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

8
**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

9
**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 – 40x x. 4x3 – 40x x 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 – 40x x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Factor Completely

10
**Factoring Polynomials of the Form ax2 + bx + c**

Factoring complex trinomials of the form ax2 + bx + c, (a 1) can be done by decomposition or cross-check method. Example: Factor 3x2 + 8x + 4. 3 4 = 12 Decomposition Method 2. We need to find factors of 12 1, 12 2, 6 3, 4 1. Find the product of first and last terms is 8 whose sum 3. Rewrite the middle term decomposed into the two numbers 3x2 + 2x + 6x + 4 = (3x2 + 2x) + (6x + 4) 4. Factor by grouping in pairs = x(3x + 2) + 2(3x + 2) = (3x + 2) (x + 2) 3x2 + 8x + 4 = (3x + 2) (x + 2) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Factoring Polynomials of the Form ax2 + bx + c

11
**Example: Factor 4x2 + 8x – 5. 4 5 = 20 1, 20 2, 10 4, 5**

We need to find factors of 20 1, 20 2, 10 4, 5 is 8 whose difference Rewrite the middle term decomposed into the two numbers 4x2 – 2x + 10x – 5 = (4x2 – 2x) + (10x – 5) Factor by grouping in pairs = 2x(2x – 1) + 5(2x – 1) = (2x – 1) (2x + 5) 4x2 + 8x – 5 = (2x –1)(2x – 5) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Factor

Similar presentations

OK

Factoring Checklist Works every time!. 1. Check to see if there is a GCF. If so, factor it out. 3xy² + 12xy.

Factoring Checklist Works every time!. 1. Check to see if there is a GCF. If so, factor it out. 3xy² + 12xy.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google

View my ppt online shopping Ppt on instrument landing system receiver Addition for kids ppt on batteries Ppt on power quality and energy management Ppt on first conditional worksheet Ppt on wings of fire Ppt on culture and science in the ancient period of philosophy Ppt on project tiger in india Ppt on the art of war sun Ppt on summary writing skills