Factor each of the following expressions.
1) 4x + 24y 2) 2x2 – 8x 3) x2 – 3x 4) 3a + 6ab + 9a2 5) ab + bc - 2b 6) 3(x – 6) + d(x – 6) 7) 2x(x + 4) + 3x + 12 8) 4x2y3 + 6x3y2 9) x2 + 5x + 6 10) x2 - 15x + 56 11) x2 + 2x - 15 |
12) x2 - x - 20 13) 2x2 + 9x + 9 14) 2x2 + 9x + 9 15) x2 + 8x + 16 16) x2 + 6xy + 9y2 17) 4a2 - 20ab + 25b2 18) 9m2 - 16y2 19) a2b2 - c2d2 20) x4 - y4 21) x3 + 3x2 + 6x + 18 22) x3 + 2x2 - 9x - 18 |
1. Answer :
4x + 24y
Greatest common divisor of 4x and 24y is 4.
Divide 4x and 24y by 4. The results are x and 6y.
4x + 24y = 4(x + 6y)
2. Answer :
2x2 – 8x
GCD of 2x2 and 8x is 2x.
Divide 2x2 and 8x by 2x. The results x and 4.
2x2 – 8x = 2x(x - 4)
3. Answer :
x2 – 3x
GCD of x2 and 3x is x.
Divide x2 and 3x by x. The results x and 3.
x2 – 3x = x(x - 3)
4. Answer :
3a + 6ab + 9a2
GCD of 3a, 6ab and 9a2 is 1, 2b and 3a.
3a + 6ab + 9a2 = 3a(1 + 2b + 3a)
5. Answer :
ab + bc - 2b
GCD of ab, bc and 2b is b.
Divide ab, bc and 2b by b. The results are a, c and -2.
ab + bc - 2b = b(a + c – 2)
6. Answer :
3(x – 6) + d(x – 6)
GCD of 3(x – 6) and d(x – 6) is (x - 6).
Divide 3(x – 6) and d(x – 6) by (x - 6). The results are 3 and d.
3(x – 6) + d(x – 6) = (x – 6)(3 + d)
7. Answer :
2x(x + 4) + 3x + 12
In the expression above, GCD of 3x and 12 is 3.
Divide 3x and 12 by 3. The results are x and 4.
2x(x + 4) + 3x + 12 = 2x(x + 4) + 3(x + 4)
GCD of 2x(x + 4) and 3(x + 4) is (x + 4).
Divide 2x(x + 4) and 3(x + 4) by (x + 4). The results are 2x and 3.
2x(x + 4) + 3x + 12 = (x + 4)(2x + 3)
8. Answer :
4x2y3 + 6x3y2
GCD of 4x2y3 + 6x3y2 is 2x2y2.
Divide 4x2y3 and 6x3y2 is 2x2y2. The results are 2y and 3x.
4x2y3 + 6x3y2 = 2xy(2x + 3y)
9. Answer :
x2 + 5x + 6
Find two factors of +6 such that the product is +6 and sum is +5.
Then the two factors of +6 are +2 and +3.
Split 5x into two terms using the factors +2 and +3.
x2 + 5x + 6 = x2 + 2x + 3x + 6
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3)
10. Answer :
x2 - 15x + 56
Find two factors of +56 such that the product is 56 and sum is -15.
Then the two factors of -15 are -7 and -8.
Split -15x into two terms using the factors -7 and -8.
x2 - 15x + 56 = x2 - 7x - 8x + 56
= x(x - 7) - 8(x - 7)
= (x - 7)(x - 8)
11. Answer :
x2 + 2x - 15
Find two factors of -15 such that the product is -15 and sum is +2.
Then the two factors of -15 are -3 and +5.
Split -2x into two terms using the factors -3 and +5.
x2 + 2x - 15 = x2 - 3x + 5x - 15
= x(x - 3) + 5(x - 3)
= (x - 3)(x + 5)
12. Answer :
x2 - x - 20
Find two factors of -20 such that the product is -20 and sum is -1.
Then the two factors of -20 are -5 and +4.
Split -x into two terms using the factors -5 and +4.
x2 - x - 20 = x2 - 5x + 4x - 20
= x(x - 5) + 4(x - 5)
= (x - 5)(x + 4)
13. Answer :
2x2 + 9x + 9
The product of 2 and 9 is +18.
Find two factors of +18 such that the product is +18 and sum is +9.
Then the two factors of +18 are +3 and +6.
Split 9x into two terms using the factors +3 and +6.
2x2 + 9x + 9 = 2x2 + 3x + 6x + 9
= x(2x + 3) + 3(2x + 3)
= (2x + 3)(x + 3)
14. Answer :
3x2 + x - 4
The product of 3 and -4 is -12.
Find two factors of -12 such that the product is -12 and sum is +1.
Then the two factors of -12 are -3 and +4.
Split +x into two terms using the factors -3 and +4.
3x2 + x - 4 = 3x2 - 3x + 4x - 4
= 3x(x - 1) + 4(x - 1)
= (x - 1)(3x + 4)
x2 + 8x + 16
15. Answer :
x2 + 8x + 16 = x2 + 2(x)(4) + 42
= (x + 4)2
= (x + 4)(x + 4)
16. Answer :
x2 + 6xy + 9y2 = x2 + 2(x)(3y) + (3y)2
= (x + 3y)2
= (x + 3y)(x + 3y)
17. Answer :
4a2 - 20ab + 25b2 = 22a2 - 20ab + 52b2
= (2a)2 - 20ab + (5b)2
= (2a)2 - 2(2a)(5b) + (5b)2
= (2a - 5b)2
= (2a - 5b)(2a - 5b)
18. Answer :
9m2 - 16y2 = 32m2 - 42n2
= (3m)2 - (4n2)
= (3m + 4n)(3m - 4n)
19. Answer :
a2b2 - c2d2 = (ab)2 - (cd)2
= (ab + cd)(ab - cd)
20. Answer :
x4 - y4 = (x2)2 - (y2)2
Let a = x2 and b = y2.
= a2 - b2
= (a + b)(a - b)
Substitute a = x2 and b = y2.
= (x2 + y2)(x2 - y2)
= (x2 + y2)(x + y)(x - y)
21. Answer :
x3 + 3x2 + 6x + 18
GCD of x3 and 3x2 is x2.
Divide x3 and 3x2 by x2. The results are x and 3.
Similarly GCD of 6x and 18 is 6
Divide 6x and 18 by 6. The results are x and 3.
x3 + 3x2 + 6x + 18 = x2(x + 3) + 6(x + 3)
= (x + 3)(x2 + 6)
22. Answer :
x3 + 2x2 - 9x - 18
GCD of x3 and 2x2 is x2.
Divide x3 and 2x2 by x2. The results are x and 2.
Similarly GCD of -9x and -18 is -9
Divide -9x and -18 by -9. The results are x and 2.
x3 + 2x2 - 9x - 18 = x2(x + 2) - 9(x + 2)
= (x + 2)(x2 - 9)
= (x + 2)(x2 - 32)
= (x + 2)(x + 3)(x - 3)
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