1. CONFIDENTIAL 1 Algebra I Choosing a Factoring Method

2. CONFIDENTIAL 2 Warm Up 1) x 2 - 4x + 4 2) x 2 - 4x - 4 3) 4x 2 - 12 4) 16x 2 - 225 Determining whether the trinomial/binomial is a perfect square. If so, factor. If not, explain:

3. CONFIDENTIAL 3 A trinomial is a perfect square if:  The first and the last terms are perfect squares.  The middle term is two times one factor from the first term and one factor from the last term. 9x 2 + 12x + 4 3x.3x2.22(3x.2) Perfect square trinomialsExamples a 2 + 2ab + b 2 = (a + b) (a + b) = (a + b) 2 x 2 + 6x + 9 = (x + 3) (x + 3) = (x + 3) 2 a 2 - 2ab + b 2 = (a - b) (a - b) = (a - b) 2 x 2 - 6x + 9 = (x - 3) (x - 3) = (x - 3) 2 Let’s review what we did in the last session

4. CONFIDENTIAL 4 perfect square Recognizing and factoring perfect square trinomials Determining whether the trinomial is a perfect square. If so, factor. If not, explain: 1) x 2 + 12x + 36 x 2 + 12x + 36 x.x6.62(x. 6) The trinomial is a perfect square. METHOD 1: Use the rule. x 2 + 12x + 36 = x 2 + 2(x.6) + (6) 2 a = x; b = 6 Write the trinomial as a 2 + 2ab + b 2 = (x + 6) 2 Write the trinomial as (a + b) 2 Next page

5. CONFIDENTIAL 5 METHOD 2: Factor. x 2 + 12x + 36 Factors of 36 sum 1 and 36 2 and 18 3 and 12 4 and 9 6 and 6 25 14 11 10 12 = (x + 6) 2 (x + 6)(x + 6)

6. CONFIDENTIAL 6 2) x 2 + 9x + 16 x 2 + 9x + 16 x.x4.42(x. 4) 2(x. 4) = 9x x 2 + 9x + 16 is not a perfect square because 2(x. 4) = 9x.

7. CONFIDENTIAL 7 The difference of two squares (a 2 - b 2 ) can be written as the product (a + b) (a - b). You can use this pattern to factor some polynomials. A polynomial is a difference of two squares if:  There are two terms, one subtracted from the other.  Both terms are perfect squares. 4x 2 - 9 2x · 2x3 · 3 DIFFERENCE OF TWO SQUARESEXAMPLE a 2 - b 2 = (a + b) (a - b)x 2 - 9 = (x + 3) (x - 3) (a 2 - b 2 )

8. CONFIDENTIAL 8 Recognizing and Factoring the Difference of Two Squares Determine whether each binomial is a difference of two squares. If so, factor. If not, explain. 1) x 6 - 7y 2 x 2 - 81 x · x9 · 9 The polynomial is a difference of two squares. x 2 - 9 2 a = x, b = 9 = (x + 9)(x - 9) Write the polynomial as (a + b) (a - b). x 2 - 81 = (x + 9) (x - 9)

9. CONFIDENTIAL 9 2) x 2 - 7y 2 x 6 - 7y 2 x 3 · x 3 7y 2 is not a perfect square. x 6 - 7y 2 is not the difference of two squares because 7y 2 is not a perfect square.

10. CONFIDENTIAL 10 Let’s start Solving an equation that involves that polynomial may require factoring the polynomial. A polynomial is in its fully factored form when it is written as a product that cannot be factored further. Determining Whether a Polynomial Is Completely Factored Tell whether the polynomial (2x + 6) (x + 5) is completely factored. If not, factor it. (2x + 6) (x + 5) =2 (x + 3) (x + 5) 2x + 6 can be further factored. 2 (x + 3) (x + 5) is completely factored. Factor out 2, the GCF of 2x and 6.

11. CONFIDENTIAL 11 To factor a polynomial completely, you may need to use more than one factoring method. Use the steps below 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 x 2 + bx + c, look for two numbers whose sum is b and whose product is c. To factor a x 2 + bx + c, check factors of a and factors of 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.

12. CONFIDENTIAL 12 Now you try! 1) 5x 2 (x - 1) 2) (4x + 4) (x + 1) Tell whether the polynomial is completely factored. If not, factor it.

13. CONFIDENTIAL 13 Factoring by GCF and Recognizing Patterns Factor -2xy 2 + 16xy - 32x completely. Check your answer. -2xy 2 + 16xy - 32x = -2x(y 2 - 8y + 16) =-2x(y - 4) 2 Factor out the GCF. y 2 - 8y + 16 is a perfect square trinomial of the form a 2 - 2ab + b 2. a = y, b = 4 Check: -2x(y - 4) 2 = -2x(y 2 - 8y + 16) -2xy 2 + 16xy - 32x If none of the factoring methods work, the polynomial is said to be unfactorable.

14. CONFIDENTIAL 14 Factor each polynomial completely. Check your answer. Now you try! 1) 4x 3 + 16x 2 + 16x 2) 2x 2 y - 2y 3

15. CONFIDENTIAL 15 Factoring by Multiple Methods Factor each polynomial completely. 1) 2x 2 + 5x + 4 ( x + )( x + ) The GCF is 1 and there is no pattern. a = 2 and c = 4; outer + inner = 5. Factors of 2 Factors of 4 outer + inner 1 and 2 1 and 4 4 and 1 2 and 2 1(4) + 2(1) = 6 1(1) + 2(4) = 9 1(2) + 2(2) = 6 2x 2 + 5x + 4 is unfactorable.

16. CONFIDENTIAL 16 2) 3n 4 - 15n 3 + 12n 2 3n 2 (n 2 - 5n + 4) (x + )( x + ) Factor out the GCF. There is no pattern. b = -5 and c = 4; look for factors of 4 whose sum is -5. Factors of 4 Sum -1 and -4 -2 and -2 -5 -4 The factors needed are -1 and -4. 3n 4 - 15n 3 + 12n 2 = 3n 2 (n - 1)(n - 4)

17. CONFIDENTIAL 17 3) p 5 - p p(p 4 - 1) =p(p 2 + 1)(p 2 - 1) =p(p 2 + 1)(p + 1)(p - 1) Factor out the GCF. p 4 - 1 is a difference of two squares. p 2 - 1 is a difference of two squares.

18. CONFIDENTIAL 18 Factor each polynomial completely. Check your answer. Now you try! 1) 3x 2 + 7x + 4 2) 2p 5 + 10p 4 - 12p 3 3) 9q 6 + 30q 5 + 24q 4

19. CONFIDENTIAL 19 Methods to Factor Polynomials Any Polynomial—Look for the greatest common factor. ab - ac = a(b - c)6x 2 y + 10xy 2 = 2xy (3x + 5y) Binomials—Look for a difference of two squares. a 2 - b 2 = (a + b)(a - b)x 2 - 9y 2 = (x + 3y)(x - 3y) Trinomials—Look for perfect-square trinomials and other factorable trinomials. a 2 + 2ab + b 2 = (a + b) 2 a 2 - 2ab + b 2 = (a - b) 2 x 2 + 4x + 4 = (x + 2) 2 x 2 - 4x + 4 = (x - 2) 2

20. CONFIDENTIAL 20 Trinomials—Look for perfect-square trinomials and other factorable trinomials. x 2 + bx + c = (x + )(x + ) x 2 + 3x + 2 = (x + 1)(x + 2) 6x 2 + 7x + 2= (2x+1)(3x+2) Polynomials of Four or More Terms—Factor by grouping. ax + bx + ay + by = x(a + b) + y(a + b) = (x + y) (a + b) 2x 3 +4x 2 +x+ 2 =(2x 3 + 4x 2 ) + (x + 2) = 2x 2 (x + 2) + 1(x + 2) = (x + 2)(2x 2 + 1)

21. CONFIDENTIAL 21 BREAK

22. CONFIDENTIAL 22

23. CONFIDENTIAL 23 Assignments 1) 2x (x 2 + 4) 2) 3x(9x 2 - 1) 3) 4x 3 - 4x 2 - 8x Tell whether each polynomial is completely factored. If not, factor it:

24. CONFIDENTIAL 24 4) 4x 3 + 18x 2 + 20x 5) 2x 4 + 18 6) 3x 5 - 12x 3 7) 4x 3 + 8x 2 + 4x Factor each polynomial completely. Check your answer.

25. CONFIDENTIAL 25 8) The square of Ella’s age plus 12 times Ella’s age plus 36. Write an expression for each situation. Factor your expression. 9) The square of the distance from point A to point B minus 81. 10) Factor and simplify: (2x + 3) 2 - (x - 4) 2

26. CONFIDENTIAL 26 Solving an equation that involves that polynomial may require factoring the polynomial. A polynomial is in its fully factored form when it is written as a product that cannot be factored further. Determining Whether a Polynomial Is Completely Factored Tell whether the polynomial (2x + 6) (x + 5) is completely factored. If not, factor it. (2x + 6) (x + 5) =2 (x + 3) (x + 5) 2x + 6 can be further factored. 2 (x + 3) (x + 5) is completely factored. Factor out 2, the GCF of 2x and 6. Let’s review

27. CONFIDENTIAL 27 To factor a polynomial completely, you may need to use more than one factoring method. Use the steps below 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 x 2 + bx + c, look for two numbers whose sum is b and whose product is c. To factor a x 2 + bx + c, check factors of a and factors of 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.

28. CONFIDENTIAL 28 Factoring by GCF and Recognizing Patterns Factor -2xy 2 + 16xy - 32x completely. Check your answer. -2xy 2 + 16xy - 32x = -2x(y 2 - 8y + 16) =-2x(y - 4) 2 Factor out the GCF. y 2 - 8y + 16 is a perfect square trinomial of the form a 2 - 2ab + b 2. a = y, b = 4 Check: -2x(y - 4) 2 = -2x(y 2 - 8y + 16) -2xy 2 + 16xy - 32x If none of the factoring methods work, the polynomial is said to be unfactorable.

29. CONFIDENTIAL 29 Factoring by Multiple Methods Factor each polynomial completely. 1) 2x 2 + 5x + 4 ( x + )( x + ) The GCF is 1 and there is no pattern. a = 2 and c = 4; outer + inner = 5. Factors of 2 Factors of 4 outer + inner 1 and 2 1 and 4 4 and 1 2 and 2 1(4) + 2(1) = 6 1(1) + 2(4) = 9 1(2) + 2(2) = 6 2x 2 + 5x + 4 is unfactorable.

30. CONFIDENTIAL 30 2) 3n 4 - 15n 3 + 12n 2 3n 2 (n 2 - 5n + 4) (x + )( x + ) Factor out the GCF. There is no pattern. b = -5 and c = 4; look for factors of 4 whose sum is -5. Factors of 4 Sum -1 and -4 -2 and -2 -5 -4 The factors needed are -1 and -4. 3n 4 - 15n 3 + 12n 2 = 3n 2 (n - 1)(n - 4)

31. CONFIDENTIAL 31 3) p 5 - p p(p 4 - 1) =p(p 2 + 1)(p 2 - 1) =p(p 2 + 1)(p + 1)(p - 1) Factor out the GCF. p 4 - 1 is a difference of two squares. p 2 - 1 is a difference of two squares.

32. CONFIDENTIAL 32 Methods to Factor Polynomials Any Polynomial—Look for the greatest common factor. ab - ac = a(b - c)6x 2 y + 10xy 2 = 2xy (3x + 5y) Binomials—Look for a difference of two squares. a 2 - b 2 = (a + b)(a - b)x 2 - 9y 2 = (x + 3y)(x - 3y) Trinomials—Look for perfect-square trinomials and other factorable trinomials. a 2 + 2ab + b 2 = (a + b) 2 a 2 - 2ab + b 2 = (a - b) 2 x 2 + 4x + 4 = (x + 2) 2 x 2 - 4x + 4 = (x - 2) 2

33. CONFIDENTIAL 33 Trinomials—Look for perfect-square trinomials and other factorable trinomials. x 2 + bx + c = (x + )(x + ) x 2 + 3x + 2 = (x + 1)(x + 2) 6x 2 + 7x + 2= (2x+1)(3x+2) Polynomials of Four or More Terms—Factor by grouping. ax + bx + ay + by = x(a + b) + y(a + b) = (x + y) (a + b) 2x 3 +4x 2 +x+ 2 =(2x 3 + 4x 2 ) + (x + 2) = 2x 2 (x + 2) + 1(x + 2) = (x + 2)(2x 2 + 1)

34. CONFIDENTIAL 34 You did a great job today!