Question

practice

1. $x^{2}-x-56$
2. $y^{2}+8y+15$
3. $x^{3}-5x^{2}-24x$
4. $x^{2}-36$
5. $4m^{3}-64m$
6. $x^{2}-7x+12$
7. $6a^{2}-48a-120$
8. $y^{2}+9y-22$
9. $x^{2}-8x$

Explanation:

Step1: Factor quadratic trinomial

Find two numbers that multiply to $-56$ and add to $-1$. These numbers are $-8$ and $7$.
$x^2 - x - 56 = (x - 8)(x + 7)$

Step2: Factor quadratic trinomial

Find two numbers that multiply to $15$ and add to $8$. These numbers are $3$ and $5$.
$y^2 + 8y + 15 = (y + 3)(y + 5)$

Step3: Factor out GCF then quadratic

First factor out $x$: $x^3 - 5x^2 - 24x = x(x^2 - 5x - 24)$.
Find two numbers that multiply to $-24$ and add to $-5$: $-8$ and $3$.
$x(x^2 - 5x - 24) = x(x - 8)(x + 3)$

Step4: Factor difference of squares

Use $a^2 - b^2=(a-b)(a+b)$, where $a=x, b=6$.
$x^2 - 36 = (x - 6)(x + 6)$

Step5: Factor out GCF then difference of cubes

First factor out $4m$: $4m^3 - 64m = 4m(m^2 - 16)$.
Factor $m^2-16$ as difference of squares: $m^2-16=(m-4)(m+4)$.
$4m(m^2 - 16) = 4m(m - 4)(m + 4)$

Step6: Factor quadratic trinomial

Find two numbers that multiply to $12$ and add to $-7$. These numbers are $-3$ and $-4$.
$x^2 - 7x + 12 = (x - 3)(x - 4)$

Step7: Factor out GCF then quadratic

First factor out $6$: $6a^2 - 48a - 120 = 6(a^2 - 8a - 20)$.
Find two numbers that multiply to $-20$ and add to $-8$: $-10$ and $2$.
$6(a^2 - 8a - 20) = 6(a - 10)(a + 2)$

Step8: Factor quadratic trinomial

Find two numbers that multiply to $-22$ and add to $9$. These numbers are $11$ and $-2$.
$y^2 + 9y - 22 = (y + 11)(y - 2)$

Step9: Factor out GCF

Factor out the greatest common factor $x$.
$x^2 - 8x = x(x - 8)$

Answer:

1. $(x - 8)(x + 7)$
2. $(y + 3)(y + 5)$
3. $x(x - 8)(x + 3)$
4. $(x - 6)(x + 6)$
5. $4m(m - 4)(m + 4)$
6. $(x - 3)(x - 4)$
7. $6(a - 10)(a + 2)$
8. $(y + 11)(y - 2)$
9. $x(x - 8)$