Question

question #6 paul and joey are determining the restricted domain values and then simplifying the following rational expression. determine who has an error in their work and explain why it is incorrect. $\frac{x^{2}-14x + 48}{3x^{2}-16x - 12}$ pauls work step 1 factor numerator $\frac{(x - 8)(x - 6)}{3x^{2}-16x - 12}$ step 2 factor denominator $\frac{(x - 8)(x - 6)}{(3x + 2)(x - 6)}$ step 3 state restricted values $x=-\frac{2}{3},x = 6$ step 4 write in simplified form $\frac{x - 8}{3x+2}$ joeys work step 1 factor numerator $\frac{(x + 8)(x + 6)}{3x^{2}-16x - 12}$ step 2 factor denominator $\frac{(x + 8)(x + 6)}{(3x - 2)(x + 6)}$ step 3 state restricted values $x=\frac{2}{3},x=-6$ step 4 write in simplified form $\frac{x + 8}{3x - 2}$ paul made an error in step 4 when writing the rational expression in simplest form. paul made an error in step 2 when factoring the denominator. joey made an error in step 3 when determining the restricted values of the domain. joey made an error in step 1 when factoring the numerator.

Explanation:

Step1: Factor the numerator

For the numerator $x^{2}-14x + 48$, we need two numbers that multiply to $48$ and add up to $-14$. The numbers are $-6$ and $-8$, so $x^{2}-14x + 48=(x - 6)(x - 8)$. Paul factored the numerator correctly as $(x - 6)(x - 8)$, while Joey factored it as $(x + 8)(x + 6)$ which is incorrect.

Step2: Factor the denominator

For the denominator $3x^{2}-16x-12$, we use the AC - method. $A = 3$, $B=-16$, $C=-12$, $AC=3\times(-12)=-36$. We find two numbers that multiply to $-36$ and add up to $-16$. The numbers are $-18$ and $2$. Then $3x^{2}-16x - 12=3x^{2}-18x+2x - 12=3x(x - 6)+2(x - 6)=(3x + 2)(x - 6)$. Paul factored the denominator correctly as $(3x + 2)(x - 6)$.

Step3: Determine restricted values

The restricted values are found by setting the denominator equal to zero. For $(3x + 2)(x - 6)=0$, we have $3x+2 = 0$ gives $x=-\frac{2}{3}$ and $x - 6=0$ gives $x = 6$. Paul found the restricted values correctly.

Step4: Simplify the expression

The original expression $\frac{x^{2}-14x + 48}{3x^{2}-16x-12}=\frac{(x - 6)(x - 8)}{(3x + 2)(x - 6)}=\frac{x - 8}{3x+2}$ (for $x
eq6,x
eq-\frac{2}{3}$). Paul simplified the expression correctly.

Answer:

Joey made an error in step 1 when factoring the numerator.