Preview Warm Up Lesson Presentation

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)

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.

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.

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.

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

Check It Out! Example 1 Tell whether each expression is completely factored. If not, factor it. A. 5x2(x – 1) 5x2(x – 1) Neither 5x2 nor x – 1 can be factored further. 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.

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.

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.

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

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

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. 4x3 + 16x2 + 16x 4x(x2 + 4x + 4) 4x(x + 2)2 a = x, b = 2 Check 4x(x + 2)2 = 4x(x2 + 2x + 2x + 4) = 4x(x2 + 4x + 4) = 4x3 + 16x2 + 16x 

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 Check 2y(x + y)(x – y) = 2y(x2 – xy + xy – y2) = 2x2y – 2xy2 + 2xy2 – 2y3 = 2x2y – 2y3 

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. Helpful Hint

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 ( x + )( x + ) a = 9 and c = –2; Outer + Inner = 3  Factors of 9 Factors of 2 Outer + Inner 1 and 9 1 and –2 1(–2) + 1(9) = 7 3 and 3 3(–2) + 1(3) = –3 –1 and 2 3(2) + 3(–1) = 3  (3x – 1)(3x + 2)

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

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) (y + )(y + )  Factors of –18 Sum –1 and 18 17 –2 and 9 7  –3 and 6 3 The factors needed are –3 and 6. 4(y – 3)(y + 6)

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

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

 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) (p + )(p + )  Factors of – 6 Sum – 1 and 6 5 The factors needed are –1 and 6. 2p3(p + 6)(p – 1)

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

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.

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)