Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

Ask by Harper Cruz. in the United States

Mar 29,2025

Question

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$ \frac{y^{2}-2 y-63}{y^{2}+16 y+63} $$

UpStudy AI · Accepted Answer

1. Factor the numerator:
   $$
   y^2 - 2y - 63 = (y + 7)(y - 9)
   $$
   since $$ (y+7)(y-9) = y^2 - 9y + 7y - 63 = y^2 - 2y - 63 $$.

2. Factor the denominator:
   $$
   y^2 + 16y + 63 = (y + 7)(y + 9)
   $$
   because $$ (y+7)(y+9) = y^2 + 9y + 7y + 63 = y^2 + 16y + 63 $$.

3. Write the rational expression in its factored form:
   $$
   \frac{(y+7)(y-9)}{(y+7)(y+9)}
   $$

4. Cancel the common factor $$(y+7)$$, noting that $$y+7 \neq 0$$ (i.e., $$y \neq -7$$):
   $$
   \frac{y-9}{y+9}
   $$

5. Identify the values excluded from the domain:
   - From the original denominator $$ (y+7)(y+9) $$, we must have $$y+7 \neq 0$$ and $$y+9 \neq 0$$.
   - Thus, $$y \neq -7$$ and $$y \neq -9$$.

The simplified rational expression is:
$$
\frac{y-9}{y+9}
$$
with the domain restrictions:
$$
y \neq -7, \quad y \neq -9
$$
The simplified rational expression is $$ \frac{y-9}{y+9} $$. It is undefined when $$ y = -7 $$ or $$ y = -9 $$.

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

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