Pre-Calculus Math 11 Section 4.2S Page 7

FACTORING COMPLETELY

STEP 1: Remove common factor(s) (if possible)

STEP 2: Continue to factor using any of the methods below (if possible)

Note: If you cannot factor the expression at all write DNF - Does Not Factor.

1. Common Factor

a) 6x + 3 b) 4c2d3 – 6cd3 c) 9x – 5x2 d) 3x3y – 6xy2 +12xy

e) 5(a - 1) + b(1 - a)

2. Difference of Squares x2 – y2 = (x - y)(x + y)

a) x2 – 1 c) x4 – 16

b) x2 + 4 d) 9a2 – 49b2 Check:

3. Trinomials x2 + bx + c = c ,

+ = b

a) x2 + 5x + 6 c) x2 + 4xy - 5y2 Check:

b) x2 - 5x - 6 d) 75 – 30x + 3x2

PC 11 4.2 SUPPLEMENT: FACTORING REVIEW - EXAMPLES

“Negative Trick” 1 – a = - a + 1

= - 1 (a -1)

Binomials have 2 terms

Trinomials have 3

terms

Pre-Calculus Math 11 Section 4.2S Page 8

4. Trinomials ax2 + bx + c, a 1 This process is called “Decomposition”,

a) 4x2 - 4x - 3 (1) Find two numbers that satisfy = ac

+ = b

(2) Rewrite the polynomial. Use the two numbers to

rewrite (Decompose) the middle term

(3) Group the first two and the last two terms.

Remove common factors from each group.

(4) You now have common binomial factors (remove it)

and write the factors.

b) 6m2 + 11mn – 10n2 Check:

5. Grouping

a) b2 - 4ab + 3ab - 12a2 b) y3 - y2 - 4y + 4

6. Using Replacement

a) (x + 2)2 + 3(x + 2) + 2 b) (a + 3b)2 - (5a + 7b)2

Pre-Calculus Math 11 Section 4.2S Page 9

1. Common Factors

a) 6x + 12 f) 18m2n4 - 12m2n3 + 24m2n2

b) 24x2 - 8x g) 4a(x + y) - 5(x + y)

c) 10x2y - 15xy2 h) 2a(c - d) + 3b(d - c)

d) 12x2 - 9x + 15 i) 6x(4a+3) - 5x(4a + 3) - x(3 + 4a)

e) 2x3 - 3x2 + 5x j) 8x(a + 5) - 3(a - 5)

2. Difference of Squares

a) x2 - 1 f) -x2 + 16

b) x2 + 4 g) 25 + 4x2

c) 9y2 - 16 h) 12x2 - 75

d) 49x4 - 1 I) a2b - b3

e) x2 - 16y2 j) 4/9 x2 - 16

3. Trinomials x2 + bx + c

a) x2 + 6x + 8 h) x2 + 8x - 16

b) y2 - 9y + 14 i) 3 - 4t + t2

c) a2 - 10a + 25 j) x2 - 11xy - 60y2

d) m2 + 13m - 36 k) 15r2 + 2rs - s2

e) x2 - x - 6 l) a2 + 4ab - 60b2

f) x2 + 8x + 16 m) a4 + a2 - 56

g) 45 + 14y + y2 n) 3x2 + 21xy - 54y2

PC 11 4.2 SUPPLEMENT: FACTORING REVIEW – PART A

May want to complete on a separate sheet of

paper

Pre-Calculus Math 11 Section 4.2S Page 10

4. Trinomials ax2 + bx + c “Decomposition”

a) 2x2 - 5x - 3 f) 10x2 + 13x - 3

b) 2y2 + 15y + 7 g) 9n - 5 + 2n2

c) 2a2 + 13a + 15 h) 8x2 - 10xy + 3y2

d) 9m2 + 6m - 8 i) 2a2 + 7ab - 15b2

e) 5x2 + 2x + 7 j) 4a2 - 4ab - 5b2

5. Grouping

a) x2 + 2x + xy + 2y c) 4u2 + v + 2uv + 2u

b) 10 - 2t - 5s - st d) ad + 3a - d2 -3d

6. Using Replacement

a) (x + 1)2 + (x + 1) - 56 d) (x + 4)2 - (y + 1)2

b) (a + b)2 - 3(a + b) - 40 e) (a - y)2 - (x + y)2

c) 5(a - 1)2 - 17(1 - a) + 6 f) 4(2x + 3)2 - 81

Pre-Calculus Math 11 Section 4.2S Page 11

Show all work on this handout

Factor each question completely.

If the expression doesn’t factor write: DNF and briefly explain why.

1. 12a3b7 + 4a2b5 - 8ab2 2. x2 - 10x - 11

3. 5c3d + 70c2d2 + 60cd3 4. j2 - 169

5. 27x2 – 300y2 6. 7y(y + 3x) + 2x(3x + y)

7. n2 – 13np + 30p2 8. (a + 5x)2 - 49x2

9. 5a(a – 4b) + 7b(4b – a) 10. 6x3 + 3x2 - 10x - 5

11. 3x2 + 5x – 2 12. k4 - 5k2 - 6

PC 11 4.2 SUPPLEMENT: FACTORING REVIEW – PART B

Pre-Calculus Math 11 Section 4.2S Page 12

13. 18x4y3 + 9x3y4 - 3x3y3 14. 12a2 - 7ab + b2

15. (x + y)2 + 9(x + y) + 20 16. 81b2 - 64a2

17. (3w – 5)2 – (w + 7)2 18. -3y2 + 75y + 162

19. d2 + 6d + 9 20. n2 + 4

21. 45x3y + 35x2y2 - 10xy3 22. 2x4 + 10x2 + 72

23. 9x2 – 1/4y2 24. x2 + x + 20

25. 16x4 - 1 26. 3m3 + 21m2n + 4m + 28n

Show how you could check your answers for:

#11 #16

Pre-Calculus Math 11 Section 4.2S Page 13

4.2S SUPPLEMENT: FACTORING REVIEW – PART A - Answers

1a) 6(x+2) f) 6m2n2(3n2-2n+4)

b) 8x(3x-1) g) (x+y)(4a-5)

c) 5xy(2x – 3y) h) (c-d)(2a+3b)

d) 3(4x2-3x+5) i) (4a+3)(6x-5x-x)= 0

e) x(2x2-3x+5) j) DNF

2a) (x+1)(x-1) f)-(x+4)(x-4)

b) DNF g) DNF

c) (3y+4)(3y-4) h) 3(2x+5)(2x-5)

d) (7x2+1) )(7x2-1) i) b(a+b)(a-b)

e) (x+4y)(x-4y) j) (

x+4)(

x -4)

3a) (x+2)(x+4) h) DNF

b) (y-7)(y-2) i) (t-1)(t-3)

c) (a-5)2 j) (x-15y)(x+4y)

d) DNF k) -(s-5r)(s + 3r)

e) (x-3)(x+2) l) (a-6b)(a+10b)

f) (x+4)2 m) (a2+8)(a2-7)

g) (y+9)(y+5) n) 3(x+9y)(x-2y)

4a) (x-3)(2x+1) f) (2x+3)(5x-1)

b) (y+7)(2y+1) g) (n+5)(2n-1)

c) (a+5)(2a+3) h) (4x-3y)(2x-y)

d) (3m+4)(3m-2) i) (a+5b)(2a-3b)

e) DNF j) DNF

5a) (x+z)(x +y) c) (2u+1)(2u+v)

b) DNF d) (d+3)(a-d)

6a) (x+9)(x-6) d) (x+y+5)(x-y+3)

b) (a+b-8)(a+b+5) e) (a+x)(a-x-2y)

c) (a-4)(5a-7) f) (4x+15)(4x-3)

4.2S SUPPLEMENT: FACTORING REVIEW – PART B - Answers

1. 4ab2(3a2b5 + b3 - 2) 2. (x – 11)(x + 11)

3. 5cd(c2 + 14cd + 12d2) 4. (j - 13)(j + 13)

5. 3(3x – 10y)(3x + 10y) 6. (3x + y)(7y + 2x)

7. (n – 3p)(n – 10p) 8. (a - 2x)(a + 12x)

9. (a – 4b)(5a – 7b) 10. (2x + 1)(3x2 – 5)

11. (x + 2)(3x –1) 12. (k2 – 6)(k2 + 1)

13. 3x3y3(6x + 3y – 1) 14. (3a – b)(4a – b)

15. (x + y + 4)(x + y + 5) 16. (9b – 8a)(9b + 8a)

17. 4(w – 6)(2w + 1) 18. -3(y – 27)(y + 2)

19. (d + 3)2 20. DNF

21. 5xy(x + y)(9x – 2y 22. 2(x2 + 9)(x + 2)(x – 2)

23. (3x – ½y)(3x + ½y) 24. DNF

25. (2x – 1)(2x + 1)(4x2 + 1) 26. (m + 7n)(3m2 + 4)