To determine whether the given rational expression [tex]\(\frac{4x^2 + 15}{4x^2 - 9}\)[/tex] is proper or improper, we need to compare the degrees of the numerator and the denominator.
1. Identify the degrees:
- The degree of the numerator [tex]\(4x^2 + 15\)[/tex] is 2.
- The degree of the denominator [tex]\(4x^2 - 9\)[/tex] is 2.
2. Because the degrees of the numerator and denominator are the same (both are 2), the rational expression [tex]\(\frac{4x^2 + 15}{4x^2 - 9}\)[/tex] is classified as an improper fraction.
Since it's an improper fraction, we need to rewrite it as the sum of a polynomial and a proper rational expression. We do this by performing polynomial division:
3. Polynomial Division:
- Divide [tex]\(4x^2 + 15\)[/tex] (numerator) by [tex]\(4x^2 - 9\)[/tex] (denominator).
Performing the division, we get:
[tex]\[
4x^2 + 15 \div 4x^2 - 9 = 1 + \frac{24}{4x^2 - 9}
\][/tex]
Here:
- [tex]\(1\)[/tex] is the quotient.
- [tex]\(24/(4x^2 - 9)\)[/tex] is the remainder over the original denominator.
Therefore, the rewritten form of the rational expression is:
[tex]\[
\frac{4x^2 + 15}{4x^2 - 9} = 1 + \frac{24}{4x^2 - 9}
\][/tex]
So, the correct choice is:
A. The expression is improper; [tex]\(\frac{4 x^2+15}{4 x^2-9}=\)[/tex] [tex]\(1 + \frac{24}{4x^2 - 9}\)[/tex].
