To simplify [tex]\(\left(15 x^2 - 24 x + 9\right) \div (3 x - 3)\)[/tex], we need to perform polynomial division.
### Step-by-Step Solution
1. Set up the division: We are dividing the polynomial [tex]\(15 x^2 - 24 x + 9\)[/tex] by [tex]\(3 x - 3\)[/tex].
2. Divide the leading terms:
- The leading term of the numerator is [tex]\(15 x^2\)[/tex].
- The leading term of the denominator is [tex]\(3 x\)[/tex].
- Divide [tex]\(15 x^2\)[/tex] by [tex]\(3 x\)[/tex]: [tex]\(\frac{15 x^2}{3 x} = 5 x\)[/tex].
This tells us that the leading term of the quotient is [tex]\(5 x\)[/tex].
3. Multiply the entire denominator by this term:
- Multiply [tex]\(3 x - 3\)[/tex] by [tex]\(5 x\)[/tex]:
[tex]\[
(3 x)(5 x) + (-3)(5 x) = 15 x^2 - 15 x
\][/tex]
4. Subtract this result from the original numerator:
[tex]\[
(15 x^2 - 24 x + 9) - (15 x^2 - 15 x) = (-24 x + 15 x) + 9 = -9 x + 9
\][/tex]
5. Repeat the process with the new polynomial:
- Divide the leading term of the new polynomial [tex]\(-9 x\)[/tex] by the leading term of the denominator [tex]\(3 x\)[/tex]:
[tex]\[
\frac{-9 x}{3 x} = -3
\][/tex]
This tells us that the next term of the quotient is [tex]\(-3\)[/tex].
6. Multiply the entire denominator by this new term:
- Multiply [tex]\(3 x - 3\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[
(3 x)(-3) + (-3)(-3) = -9 x + 9
\][/tex]
7. Subtract this result from the current polynomial:
[tex]\[
(-9 x + 9) - (-9 x + 9) = 0
\][/tex]
Since the remainder is zero, the division is exact.
### Final Result
The quotient of the division is:
[tex]\[
5 x - 3
\][/tex]
The simplified form of [tex]\(\left(15 x^2 - 24 x + 9\right) \div (3 x - 3)\)[/tex] is therefore:
[tex]\[
5 x - 3
\][/tex]
So the correct answer is:
D. [tex]\(5 x - 3\)[/tex]
