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Example 1: Determining Whether a Polynomial is Completely Factored

Recall that a polynomial is in its fully factored form when it is written as a product that cannot be factored further. Example 1: Determining Whether a Polynomial is Completely Factored. Tell whether each polynomial is completely factored. If not factor it. A. 3 x 2 (6 x – 4).

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Example 1: Determining Whether a Polynomial is Completely Factored

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  1. Recall that a polynomial is in its fully factored form when it is written as a product that cannot be factored further.

  2. Example 1: Determining Whether a Polynomial is Completely Factored Tell whether each polynomial is completely factored. If not factor it. A. 3x2(6x– 4) 3x2(6x–4) 6x – 4 can be further factored. 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.

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

  4. Check It Out! Example 1 Tell whether the polynomial 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.

  5. To factor a polynomial completely, you may need to use more than one factoring method. Use the steps below to factor a polynomial completely.

  6. = 10x2 + 48x + 32 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 remaining trinomial. Check 2(5x + 4)(x + 4) = 2(5x2 + 20x + 4x + 16) = 10x2 + 40x + 8x + 32

  7. = 8x6y2– 18x2y2 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 = 2x, b = 3 Check 2x2y2(2x2– 3)(2x2 + 3) = 2x2y2(4x4– 9)

  8. 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 

  9. 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

  10. Helpful Hint For a polynomial of the form ax2 + bx + c, if there are no numbers whose sum is b and whose product is ac, then the polynomial is unfactorable. If none of the factoring methods work, the polynomial is said to be unfactorable.

  11. ( 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) 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

  12. (x + )(x + ) Factors of 4Sum  1 and 4 5  2 and 2 4 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 c = 2 12b(b + 2)(b + 2) 12b(b + 2)2

  13. (y + )(y + ) Factors of –18Sum  –1 and 18 17  –2 and 9 7  –3 and 6 3 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)

  14. 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)

  15. ( 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

  16. (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)

  17. ( 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

  18. 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.

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