1 / 25

Polynomial Factoring Techniques: California Standards Lesson

In this lesson, students will learn basic factoring techniques for second- and simple third-degree polynomials. The lesson includes examples of factoring trinomials, perfect-square trinomials, and using the difference of two squares. Additionally, students will learn how to determine if an expression is completely factored and solve practice problems.

zapata
Download Presentation

Polynomial Factoring Techniques: California Standards Lesson

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Preview Warm Up California Standards Lesson Presentation

  2. Warm Up • Factor each trinomial. • 1. x2 + 13x + 40 (x + 5)(x + 8) 2. 5x2– 18x– 8 (5x + 2)(x– 4) 3. Factor the perfect-square trinomial 16x2 + 40x + 25. (4x + 5)(4x + 5) 4. Factor 9x2 – 25y2 using the difference of two squares. (3x + 5y)(3x– 5y)

  3. California Standards 11.0 Students apply basic factoring techniques to second- and simple third- degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

  4. Recall that a polynomial is fully or completely factored when it is written as a product of monomials and polynomials whose terms have no common factors other than 1.

  5. Additional Example 1: Determining Whether an Expression is Completely Factored Tell whether each expression is completely factored. If not, factor it. A. 3x2(6x– 4) 3x2(6x– 4) 6x – 4 can be factored further. 6x2(3x– 2) Factor out 2, the GCF of 6x and – 4. 6x2(3x– 2) is completely factored. B. (x2 + 1)(x– 5) (x2 + 1)(x– 5) Neither x2 +1 nor x – 5 can be factored further. (x2 + 1)(x– 5) is completely factored.

  6. Additional Example 1: Determining Whether an Expression is Completely Factored Tell whether each expression is completely factored. If not, factor it. C. 5x(x2– 2x– 3) (x2 – 2x – 3) can be factored further. 5x(x2– 2x– 3) 5x(x + 1)(x– 3) Factor x2 – 2x – 3. 5x(x + 1)(x– 3) is completely factored.

  7. Caution x2 + 4 is a sum of squares, and cannot be factored.

  8. Check It Out! Example 1 Tell whether each expression is completely factored. If not, factor it. A. 5x2(x– 1) Neither 5x2 nor x – 1 can be factored further. 5x2(x– 1) 5x2(x– 1) is completely factored. B. (4x + 4)(x + 1) (4x +4)(x + 1) 4x + 4 can be further factored. 4(x +1)(x + 1) Factor out 4, the GCF of 4x and 4. 4(x + 1)2 is completely factored.

  9. To factor a polynomial completely, you may need to use more than one factoring method. Use the steps on the next slide to factor a polynomial completely.

  10. Factoring Polynomials Step 1 Check for a greatest common factor. Step 2 Check for a pattern that fits the difference of two squares or a perfect-square trinomial. Step 3 To factor x2 + bx + c, look for two integers whose sum is b and whose product is c. To factor ax2 + bx + c, check integer factors of a and c in the binomial factors. The sum of the products of the outer and inner terms should be b. Step 4 Check for common factors.

  11. = 10x2 + 48x + 32 Additional Example 2A: Factoring by GCF and Recognizing Patterns Factor 10x2 + 48x + 32 completely. Check your answer. 10x2 + 48x + 32 Factor out the GCF. 2(5x2 + 24x + 16) 2(5x + 4)(x + 4) Factor 5x2 + 24x + 16. Check 2(5x + 4)(x + 4) = 2(5x2 + 20x + 4x + 16) = 10x2 + 40x + 8x + 32

  12. = 8x6y2– 18x2y2 Additional Example 2B: Factoring by GCF and Recognizing Patterns Factor 8x6y2– 18x2y2 completely. Check your answer. 8x6y2– 18x2y2 Factor out the GCF. 4x4 – 9is a perfect-square trinomial of the form a2 – b2. 2x2y2(4x4– 9) 2x2y2(2x2– 3)(2x2 + 3) a = 2x2, b = 3 Check2x2y2(2x2– 3)(2x2 + 3) = 2x2y2(4x4– 9)

  13. 4x3 + 16x2 + 16x 4x(x2 + 4x + 4) Check It Out! Example 2a Factor each polynomial completely. Check your answer. 4x3 + 16x2 + 16x Factor out the GCF. x2 + 4x + 4 is a perfect-square trinomial of the form a2 + 2ab + b2. 4x(x + 2)2 a = x, b = 2 Check4x(x + 2)2 = 4x(x2 + 2x + 2x + 4) = 4x(x2+ 4x + 4) = 4x3 + 16x2 + 16x 

  14. Check It Out! Example 2b Factor each polynomial completely. Check your answer. 2x2y– 2y3 Factor out the GCF. 2y(x2 – y2) is a perfect-square trinomial of the form a2 – b2. 2x2y– 2y3 2y(x2–y2) 2y(x + y)(x–y) a = x, b = y Check2y(x +y)(x–y) = 2y(x2–xy + xy –y2) = 2x2y– 2xy2 + 2xy2 – 2y3  = 2x2y–2y3

  15. Helpful Hint For a polynomial of the form ax2 + bx + c, if there are no integers whose sum is b and whose product is ac, then the polynomial is said to be unfactorable.

  16. ( x + )( x + ) Factors of 9 Factors of 2Outer+Inner  1 and –2 1(–2) + 1(9) = 7 1 and 9  1 and –2 3(–2) + 1(3) = –3 3 and 3  –1 and 2 3(2) + 3(–1) = 3 3 and 3 (3x– 1)(3x + 2) Additional Example 3A: Factoring by Multiple Methods Factor each polynomial completely. 9x2 + 3x– 2 The GCF is 1 and there is no pattern. 9x2 + 3x– 2 a = 9 and c = –2; Outer + Inner = 3

  17. (b + )(b + ) Factors of 4Sum  1 and 4 5  2 and 2 4 Additional Example 3B: Factoring by Multiple Methods Factor each polynomial completely. 12b3 + 48b2 + 48b The GCF is 12b; (b2 + 4b + 4) is a perfect-square trinomial in the form of a2 + 2ab + b2. 12b(b2 + 4b + 4) a = 2 and b = 2 12b(b + 2)(b + 2) 12b(b + 2)2

  18. (y + )(y + ) Factors of –18Sum  –1 and 18 17  –2 and 9 7  –3 and 6 3 Additional Example 3C: Factoring by Multiple Methods Factor each polynomial completely. 4y2 + 12y– 72 Factor out the GCF. There is no pattern. b = 3 and c = –18; look for factors of –18 whose sum is 3. 4(y2 + 3y– 18) The factors needed are –3 and 6. 4(y –3)(y + 6)

  19. Additional Example 3D: Factoring by Multiple Methods. Factor each polynomial completely. (x4–x2) Factor out the GCF. x2(x2– 1) x2 – 1is a difference of two squares. x2(x + 1)(x– 1)

  20. ( x + )( x + ) Factors of 3 Factors of 4Outer+Inner  1 and 4 3(4) + 1(1) = 13 3 and 1  2 and 2 3(2) + 1(2) = 8 3 and 1  4 and 1 3(1) + 1(4) = 7 3 and 1 (3x + 4)(x + 1) Check It Out! Example 3a Factor each polynomial completely. 3x2 + 7x + 4 a = 3 and c = 4; Outer + Inner = 7 3x2 + 7x + 4

  21. (p + )(p + ) Factors of – 6 Sum  – 1 and 6 5 Check It Out! Example 3b Factor each polynomial completely. 2p5 + 10p4– 12p3 Factor out the GCF. There is no pattern. b = 5 and c = –6; look for factors of –6 whose sum is 5. 2p3(p2 + 5p– 6) The factors needed are –1 and 6. 2p3(p + 6)(p– 1)

  22. ( q + )( q + ) Factors of 3 Factors of 8Outer+Inner  1 and 8 3(8) + 1(1) = 25 3 and 1  2 and 4 3(4) + 1(2) = 14 3 and 1  4 and 2 3(2) + 1(4) = 10 3 and 1 3q4(3q + 4)(q + 2) Check It Out! Example 3c Factor each polynomial completely. 9q6 + 30q5 + 24q4 Factor out the GCF. There is no pattern. 3q4(3q2 + 10q + 8) a = 3 and c = 8; Outer + Inner = 10

  23. Check It Out! Example 3d Factor each polynomial completely. 2x4 + 18 2(x4 + 9) Factor out the GFC. x4 + 9 is the sum of squares and that is not factorable. 2(x4 + 9) is completely factored.

  24. Lesson Quiz Tell whether the polynomial is completely factored. If not, factor it. 1. (x + 3)(5x + 10) 2. 3x2(x2 + 9) no; 5(x+ 3)(x + 2) completely factored Factor each polynomial completely. Check your answer. 3.x3 + 4x2 + 3x + 12 4. 4x2 + 16x – 48 4(x + 6)(x– 2) (x + 4)(x2 + 3) 5. 18x2– 3x– 3 6. 18x2– 50y2 3(3x + 1)(2x– 1) 2(3x + 5y)(3x– 5y) 7. 5x –20x3 + 7 – 28x2 (1 + 2x)(1 – 2x)(5x + 7)

More Related