Question

factoring practice

1. $6x^3 - 12x$
2. $5x^2 - 5y^2$
3. $x^2 + xy - 12y^2$
4. $18x^2 - 25x - 3$
5. $x^4 - 64$
6. $3x - 9xy$
7. $x^2 + 9x + 20$
8. $-4 + 25x^2$
9. $9x^4 - 16y^2$
10. $3y^2 - 9y^3$
11. $9x^2 + 1$
12. $2x^2 - 4xy + 8x$
13. $m^4 - 6m^2 - 27$
14. $4x^2 + 24x + 36$
15. $9y^2 - 30y + 25$
16. $-75 + 12x^4$
17. $y^4 - 17y^2 + 16$
18. $x^4 + 4x - 21$
19. $3x^5 + 15x^3 + 12x$
20. $3x^2 yz^3 + 18xy^2$
21. $6xy + 3x - 4y - 2$
22. $z^4 - 20z^2 + 100$
23. $m^2 + 6m + 9 - 4n^2$
24. $16y^2 - a^2 - 6ab - 9b^2$
25. $m^4 + 3m^2 + 4$

Explanation:

Step1: Factor out GCF (Q1)

$6x^3 - 12x = 6x(x^2 - 2)$

Step2: Factor difference of squares (Q2)

$5x^2 - 5y^2 = 5(x^2 - y^2) = 5(x-y)(x+y)$

Step3: Factor quadratic trinomial (Q3)

$x^2 + xy -12y^2 = (x-3y)(x+4y)$

Step4: Factor quadratic trinomial (Q4)

$18x^2 -25x -3 = (9x+1)(2x-3)$

Step5: Factor difference of squares (Q5)

$x^4 -64 = (x^2-8)(x^2+8) = (x-2\sqrt{2})(x+2\sqrt{2})(x^2+8)$

Step6: Factor out GCF (Q6)

$3x -9xy = 3x(1-3y)$

Step7: Factor quadratic trinomial (Q7)

$x^2 +9x +20 = (x+4)(x+5)$

Step8: Factor difference of squares (Q8)

$-4 +25x^2 = 25x^2 -4 = (5x-2)(5x+2)$

Step9: Factor difference of squares (Q9)

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

Step10: Factor out GCF (Q10)

$3y^5 -9y^3 = 3y^3(y^2 -3)$

Step11: State non-factorable (Q11)

$9x^2 +1$ is irreducible over reals

Step12: Factor out GCF (Q12)

$2x^2 -4xy +8x = 2x(x-2y+4)$

Step13: Substitute & factor (Q13)

Let $u=m^2$: $u^2-6u-27=(u-9)(u+3)=(m^2-9)(m^2+3)=(m-3)(m+3)(m^2+3)$

Step14: Factor perfect square (Q14)

$4x^2 +24x +36 = 4(x^2+6x+9)=4(x+3)^2$

Step15: Factor perfect square (Q15)

$9y^2 -30y +25 = (3y-5)^2$

Step16: Factor difference of squares (Q16)

$-75+12x^4=12x^4-75=3(4x^4-25)=3(2x^2-5)(2x^2+5)$

Step17: Substitute & factor (Q17)

Let $u=y^2$: $u^2-17u+16=(u-1)(u-16)=(y^2-1)(y^2-16)=(y-1)(y+1)(y-4)(y+4)$

Step18: Factor quadratic trinomial (Q18)

$x^2 +4x -21 = (x+7)(x-3)$

Step19: Factor out GCF & trinomial (Q19)

$3x^5+15x^3+12x=3x(x^4+5x^2+4)=3x(x^2+1)(x^2+4)$

Step20: Factor out GCF (Q20)

$3x^2yz^3 +18xy^2 = 3xy(xz^3 +6y)$

Step21: Factor by grouping (Q21)

$6xy+3x-4y-2=3x(2y+1)-2(2y+1)=(3x-2)(2y+1)$

Step22: Substitute & perfect square (Q22)

Let $u=z^2$: $u^2-20u+100=(u-10)^2=(z^2-10)^2$

Step23: Factor by grouping (Q23)

$m^2+6m+9-4n^2=(m+3)^2-(2n)^2=(m+3-2n)(m+3+2n)$

Step24: Factor by grouping (Q24)

$16y^2 -a^2-6ab-9b^2=16y^2-(a+3b)^2=(4y-a-3b)(4y+a+3b)$

Step25: Rewrite & factor (Q25)

$m^4+3m^2+4=m^4+4m^2+4-m^2=(m^2+2)^2-m^2=(m^2+m+2)(m^2-m+2)$

Answer:

1. $6x(x^2 - 2)$
2. $5(x-y)(x+y)$
3. $(x-3y)(x+4y)$
4. $(9x+1)(2x-3)$
5. $(x-2\sqrt{2})(x+2\sqrt{2})(x^2+8)$
6. $3x(1-3y)$
7. $(x+4)(x+5)$
8. $(5x-2)(5x+2)$
9. $(3x^2-4y)(3x^2+4y)$
10. $3y^3(y^2 - 3)$
11. Irreducible over real numbers
12. $2x(x-2y+4)$
13. $(m-3)(m+3)(m^2+3)$
14. $4(x+3)^2$
15. $(3y-5)^2$
16. $3(2x^2-5)(2x^2+5)$
17. $(y-1)(y+1)(y-4)(y+4)$
18. $(x+7)(x-3)$
19. $3x(x^2+1)(x^2+4)$
20. $3xy(xz^3 + 6y)$
21. $(3x-2)(2y+1)$
22. $(z^2-10)^2$
23. $(m+3-2n)(m+3+2n)$
24. $(4y-a-3b)(4y+a+3b)$
25. $(m^2+m+2)(m^2-m+2)$