Let's solve the problem step by step:
We need to simplify the expression [tex]\(\frac{15x^2 - 24x + 9}{3x - 3}\)[/tex].
To do this, we perform polynomial division, which involves dividing the numerator polynomial by the denominator polynomial.
1. Divide the first terms:
[tex]\[
\frac{15x^2}{3x} = 5x
\][/tex]
2. Multiply the entire divisor by this result:
[tex]\[
(3x - 3) \cdot 5x = 15x^2 - 15x
\][/tex]
3. Subtract the result from the original polynomial:
[tex]\[
(15x^2 - 24x + 9) - (15x^2 - 15x) = -24x + 15x + 9 = -9x + 9
\][/tex]
4. Divide the remaining first term by the first term of the divisor:
[tex]\[
\frac{-9x}{3x} = -3
\][/tex]
5. Multiply the entire divisor by this new quotient term:
[tex]\[
(3x - 3) \cdot (-3) = -9x + 9
\][/tex]
6. Subtract again:
[tex]\[
(-9x + 9) - (-9x + 9) = 0
\][/tex]
So the remainder is 0, and the quotient is [tex]\(5x - 3\)[/tex].
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{15x^2 - 24x + 9}{3x - 3} = 5x - 3
\][/tex]
The correct choice is:
B. [tex]\(5x - 3\)[/tex]
