To simplify the polynomial division [tex]\(\left(15 x^2 - 24 x + 9\right) \div (3 x - 3)\)[/tex], we follow these steps:
1. Identify the polynomials: We have the numerator [tex]\(15 x^2 - 24 x + 9\)[/tex] and the denominator [tex]\(3 x - 3\)[/tex].
2. Rewrite the denominator: Notice that [tex]\(3x - 3\)[/tex] can be factored out as [tex]\(3(x - 1)\)[/tex]. This factor simplifies the division process.
3. Perform polynomial division:
- Divide the leading term of the numerator by the leading term of the denominator.
- The leading term of the numerator is [tex]\(15 x^2\)[/tex] and the leading term of the denominator is [tex]\(3x\)[/tex].
- Dividing these gives: [tex]\(\frac{15 x^2}{3 x} = 5 x\)[/tex].
4. Multiply and subtract:
- Multiply the entire denominator [tex]\(3 x - 3\)[/tex] by [tex]\(5x\)[/tex]:
[tex]\[ 5x \cdot (3x - 3) = 15 x^2 - 15 x \][/tex]
- Subtract this result from the original numerator:
[tex]\[ (15 x^2 - 24 x + 9) - (15 x^2 - 15 x) = -24 x + 9 - (- 15 x) = -24 x + 15 x + 9 = -9x + 9 \][/tex]
5. Continue the division process:
- Next, take [tex]\(-9x + 9\)[/tex] and divide by [tex]\(3x - 3\)[/tex]:
- The leading term of the new "numerator" is [tex]\(-9x\)[/tex], and the leading term of the denominator is [tex]\(3x\)[/tex].
- Dividing these gives: [tex]\(\frac{-9 x}{3 x} = -3\)[/tex].
6. Multiply and subtract again:
- Multiply the entire denominator [tex]\(3 x - 3\)[/tex] by [tex]\(-3\)[/tex]:
[tex]\[ -3 \cdot (3 x - 3) = -9 x + 9 \][/tex]
- Subtract this result from [tex]\(-9 x + 9\)[/tex]:
[tex]\[ (-9 x + 9) - (-9 x + 9) = 0 \][/tex]
7. Arrive at the quotient and remainder:
- The quotient is [tex]\(5 x - 3\)[/tex].
- The remainder is [tex]\(0\)[/tex].
8. Verify the options:
- A. [tex]\(5 x - 3\)[/tex] matches our quotient perfectly.
- Option B suggests [tex]\(5 x + 3\)[/tex], which is incorrect.
- Options C and D suggest a quotient with non-zero remainder, which does not match our result.
So, the correct answer is:
A. [tex]\(5 x - 3\)[/tex]
