Let $$ g(x)=-x^{3}+x $$. (Answered) Find and simplify the difference quotient $$ \frac{g(x)-g(a)}{x-a}, x
eq a $$. Clearly show the steps so that we can follow your solution. All fraction lines must be horizontal, not oblique. Note that factoring techniques are required (week 4 review lesson). Participation in the Math Q\&A forums is required for Weeks 3 and 4. Your replies will count toward your Class Participation (CP) grade.

Question

Let $$ g(x)=-x^{3}+x $$. (Answered) Find and simplify the difference quotient $$ \frac{g(x)-g(a)}{x-a}, x 
eq a $$. Clearly show the steps so that we can follow your solution. All fraction lines must be horizontal, not oblique. Note that factoring techniques are required (week 4 review lesson). Participation in the Math Q\&A forums is required for Weeks 3 and 4. Your replies will count toward your Class Participation (CP) grade.

UpStudy AI · Accepted Answer

1. We start with  
$$
g(x) = -x^3 + x
$$  
and need to compute the difference quotient  
$$
\frac{g(x) - g(a)}{x-a}, \quad x \neq a.
$$

2. Substitute $$ g(x) $$ and $$ g(a) $$:
$$
g(x) - g(a) = \bigl(-x^3 + x\bigr) - \bigl(-a^3 + a\bigr).
$$

3. Simplify by distributing the negative sign:
$$
g(x) - g(a) = -x^3 + x + a^3 - a.
$$

4. Rearrange the terms:
$$
g(x) - g(a) = a^3 - x^3 + x - a.
$$

5. Notice that $$ a^3 - x^3 $$ is a difference of cubes. Recall that
$$
x^3 - a^3 = (x-a)(x^2+ax+a^2).
$$
Thus,
$$
a^3 - x^3 = -(x^3-a^3) = -(x-a)(x^2+ax+a^2).
$$

6. Substitute this factorization into the expression:
$$
g(x) - g(a) = - (x-a)(x^2+ax+a^2) + (x-a).
$$

7. Factor out the common factor $$ (x-a) $$:
$$
g(x) - g(a) = (x-a) \left[ - (x^2+ax+a^2) + 1 \right].
$$

8. Now, substitute into the difference quotient:
$$
\frac{g(x) - g(a)}{x-a} = \frac{(x-a) \left[1 - (x^2+ax+a^2)\right]}{x-a}.
$$

9. Since $$ x \neq a $$, we can cancel $$ (x-a) $$:
$$
\frac{g(x) - g(a)}{x-a} = 1 - (x^2+ax+a^2).
$$

10. Finally, expand the expression inside the parentheses if desired:
$$
\frac{g(x) - g(a)}{x-a} = 1 - x^2 - ax - a^2.
$$
$$ \frac{g(x) - g(a)}{x - a} = 1 - x^2 - ax - a^2 $$

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