Question

factor these problems.

1. $5x^{2}-45=$
2. $x^{2}-8x+16=$
3. $12x^{2}-25x+12=$
4. $9a^{2}+54ab+81b^{2}=$
5. $4x^{2}-16xy+16y^{2}=$
6. $x^{2}-\frac{1}{9}=$
7. $125-x^{3}=$
8. $9x^{2}-42xy+49y^{2}=$
9. $y^{2}-81=$
10. $9a^{2}-24ab+16b^{2}=$
11. $y^{3}-8=$
12. $x^{2}-\frac{1}{4}=$
13. $x^{6}+y^{6}=$
14. $25a^{4}-16b^{2}=$
15. $x^{3}+64=$
16. $6a^{2}-5a-6=$
17. $4a^{2}-16=$
18. $4x^{2}+12x+9=$
19. $9a^{2}+6a-3=$
20. $16x^{2}-32xy+16y^{2}=$

Explanation:

Step1: Factor out GCF, use difference of squares

$5x^2 - 45 = 5(x^2 - 9) = 5(x-3)(x+3)$

Step2: Use perfect square trinomial rule

$x^2 - 8x + 16 = (x-4)^2$

Step3: Factor quadratic by grouping

$12x^2 -25x +12 = 12x^2 -16x -9x +12 = 4x(3x-4)-3(3x-4)=(4x-3)(3x-4)$

Step4: Factor out GCF, perfect square

$9a^2 +54ab +81b^2 =9(a^2+6ab+9b^2)=9(a+3b)^2$

Step5: Factor out GCF, perfect square

$4x^2 -16xy +16y^2=4(x^2-4xy+4y^2)=4(x-2y)^2$

Step6: Use difference of squares

$x^2 - \frac{1}{9}=(x-\frac{1}{3})(x+\frac{1}{3})$

Step7: Use difference of cubes

$125 -x^3=(5-x)(25+5x+x^2)$

Step8: Use perfect square trinomial rule

$9x^2 -42xy +49y^2=(3x-7y)^2$

Step9: Use difference of squares

$y^2 -81=(y-9)(y+9)$

Step10: Use perfect square trinomial rule

$9a^2 -24ab +16b^2=(3a-4b)^2$

Step11: Use difference of cubes

$y^3 -8=(y-2)(y^2+2y+4)$

Step12: Use difference of squares

$x^2 - \frac{1}{4}=(x-\frac{1}{2})(x+\frac{1}{2})$

Step13: Sum of squares, factor as sum of cubes

$x^6 + y^6=(x^2)^3+(y^2)^3=(x^2+y^2)(x^4-x^2y^2+y^4)$

Step14: Use difference of squares

$25a^4 -16b^2=(5a^2-4b)(5a^2+4b)$

Step15: Use sum of cubes

$x^3 +64=(x+4)(x^2-4x+16)$

Step16: Factor quadratic by grouping

$6a^2 -5a -6=6a^2-9a+4a-6=3a(2a-3)+2(2a-3)=(3a+2)(2a-3)$

Step17: Factor out GCF, difference of squares

$4a^2 -16=4(a^2-4)=4(a-2)(a+2)$

Step18: Use perfect square trinomial rule

$4x^2 +12x +9=(2x+3)^2$

Step19: Factor out GCF, quadratic factor

$9a^2 +6a -3=3(3a^2+2a-1)=3(3a-1)(a+1)$

Step20: Factor out GCF, perfect square

$16x^2 -32xy +16y^2=16(x^2-2xy+y^2)=16(x-y)^2$

Answer:

1. $5(x-3)(x+3)$
2. $(x-4)^2$
3. $(4x-3)(3x-4)$
4. $9(a+3b)^2$
5. $4(x-2y)^2$
6. $(x-\frac{1}{3})(x+\frac{1}{3})$
7. $(5-x)(x^2+5x+25)$
8. $(3x-7y)^2$
9. $(y-9)(y+9)$
10. $(3a-4b)^2$
11. $(y-2)(y^2+2y+4)$
12. $(x-\frac{1}{2})(x+\frac{1}{2})$
13. $(x^2+y^2)(x^4-x^2y^2+y^4)$
14. $(5a^2-4b)(5a^2+4b)$
15. $(x+4)(x^2-4x+16)$
16. $(3a+2)(2a-3)$
17. $4(a-2)(a+2)$
18. $(2x+3)^2$
19. $3(3a-1)(a+1)$
20. $16(x-y)^2$