Question

1. factor out the greatest common factor of each expression. then, check with distribution.

a) - 12a^4b + 20ab^4 - 4ab
b) - 9ym - 12yb
c) x(x - 7) - 8(x - 7)

1. fully factor: 2x(x - 7) - 6(x - 7)
2. same as one of the problems in wiley the expression 54 - 9x is equivalent to an expression of the form k(x + a). write the expression in this form and give the values of k and a.

Explanation:

a)

Step1: Identify GCF of coefficients and variables

The coefficients are -12, 20, -4. GCF of 12, 20, 4 is 4. For variables, $a^{4}b,ab^{4},ab$ have GCF $ab$. And considering the negative - sign, GCF is $-4ab$.

Step2: Divide each term by GCF

$\frac{-12a^{4}b}{-4ab}=3a^{3}$, $\frac{20ab^{4}}{-4ab}=-5b^{3}$, $\frac{-4ab}{-4ab}=1$

b)

Step1: Identify GCF of coefficients and variables

The coefficients -9 and -12 have GCF 3. The common variable is $y$. Considering the negative - sign, GCF is $-3y$.

Step2: Divide each term by GCF

$\frac{-9ym}{-3y}=3m$, $\frac{-12yb}{-3y}=4b$

c)

Step1: Notice common binomial factor

Both terms $x(x - 7)$ and $-8(x - 7)$ have the common binomial factor $(x - 7)$.

Step2: Factor out the common binomial

We get $(x - 7)(x - 8)$

2.

Step1: Factor out common factor of 2

$2x(x - 7)-6(x - 7)=2[x(x - 7)-3(x - 7)]$

Step2: Factor out common binomial $(x - 7)$

$2(x - 7)(x - 3)$

3.

Step1: Factor out -9 from $54-9x$

$54-9x=-9(-6 + x)$

Step2: Rewrite in form $k(x + A)$

$-9(x - 6)$ where $k=-9$ and $A=-6$

Answer:

a)

• Factored form: $-4ab(3a^{3}- 5b^{3}+1)$
• Check: $-4ab(3a^{3}-5b^{3}+1)=-4ab\times3a^{3}-4ab\times(- 5b^{3})-4ab\times1=-12a^{4}b + 20ab^{4}-4ab$

b)

• Factored form: $-3y(3m + 4b)$
• Check: $-3y(3m + 4b)=-3y\times3m-3y\times4b=-9ym-12yb$

c)

• Factored form: $(x - 7)(x - 8)$
• Check: $(x - 7)(x - 8)=x(x - 7)-8(x - 7)$

2.

• Factored form: $2(x - 7)(x - 3)$
• Explanation: First, factor out the common factor of 2 from $2x(x - 7)-6(x - 7)$. We get $2[x(x - 7)-3(x - 7)]$. Then factor out $(x - 7)$ to obtain $2(x - 7)(x - 3)$

3.

• Rewritten form: $-9(x - 6)$
• $k=-9$, $A=-6$
• Explanation: Factor out - 9 from $54-9x$. We have $-9(-6 + x)=-9(x - 6)$