Question

State the excluded values for each function.

$y=\frac{10}{6x^2+7x-3}$

Short answer

The excluded values are$\frac{1}{3}$and$-\frac{3}{2}$.

Step-by-step solution

Step 1. Define rational expression.

A rational expression is an algebraic expression whose numerator and denominator are polynomials. Also, division by 0 is undefined, so therefore, the polynomials in the denominator cannot be zero and the values for which the denominator is zero is termed as excluded values for the respective rational expression.

Step 2. Factor the expression.

Factor the denominator in order to find out the excluded values.

$$\begin{gathered} 6x^2+7x-3=0 \\ 6x^2+9x-2x-3=0 \\ \left(3x-1\right)\left(2x+3\right)=0 \end{gathered}$$

Step 3. Apply the zero-product rule.

Zero product rule states that if $ab=0$, then either $a=0$or $b=0$.

role="math" localid="1648046651251" $$\begin{gathered} \left(3x-1\right)\left(2x+3\right)=0 \\ 3x-1=0\text{or}2x+3=0 \\ x=\frac{1}{3}\text{or}x=-\frac{3}{2} \end{gathered}$$

Therefore, the excluded values are $-\frac{3}{2}$and $\frac{1}{3}$.