To simplify the expression [tex]\(\frac{15x^2 - 24x + 9}{3x - 3}\)[/tex], we need to perform polynomial division.
We divide the polynomial [tex]\(15x^2 - 24x + 9\)[/tex] by [tex]\(3x - 3\)[/tex].
Step-by-step procedure:
1. Setup the Division:
[tex]\[
\frac{15x^2 - 24x + 9}{3x - 3}
\][/tex]
2. Divide the Leading Terms:
Divide the leading term of the dividend [tex]\(15x^2\)[/tex] by the leading term of the divisor [tex]\(3x\)[/tex]:
[tex]\[
\frac{15x^2}{3x} = 5x
\][/tex]
3. Multiply and Subtract:
Multiply the entire divisor [tex]\(3x - 3\)[/tex] by the result from step 2:
[tex]\[
5x \cdot (3x - 3) = 15x^2 - 15x
\][/tex]
Subtract this result from the original dividend:
[tex]\[
(15x^2 - 24x + 9) - (15x^2 - 15x) = -24x + 15x + 9 = -9x + 9
\][/tex]
4. Repeat the Process:
Divide the leading term [tex]\(-9x\)[/tex] by the leading term [tex]\(3x\)[/tex]:
[tex]\[
\frac{-9x}{3x} = -3
\][/tex]
Multiply the entire divisor [tex]\(3x - 3\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
-3 \cdot (3x - 3) = -9x + 9
\][/tex]
Subtract this result from [tex]\(-9x + 9\)[/tex]:
[tex]\[
(-9x + 9) - (-9x + 9) = 0
\][/tex]
5. Quotient and Remainder:
The quotient is [tex]\(5x - 3\)[/tex] and the remainder is [tex]\(0\)[/tex].
Thus, the simplified form of the expression [tex]\(\frac{15x^2 - 24x + 9}{3x - 3}\)[/tex] is [tex]\(5x - 3\)[/tex].
Therefore, the correct answer is:
B. [tex]\(5x - 3\)[/tex]
