To divide the polynomial [tex]\(24x^4 - 24x^3 - 18x^2\)[/tex] by [tex]\(4x^3 + 2x^2\)[/tex] using long division, follow these steps:
1. Set up the division: Place the dividend [tex]\(24x^4 - 24x^3 - 18x^2\)[/tex] under the long division symbol and the divisor [tex]\(4x^3 + 2x^2\)[/tex] outside.
2. Divide the leading terms:
- The leading term of the dividend is [tex]\(24x^4\)[/tex].
- The leading term of the divisor is [tex]\(4x^3\)[/tex].
- Divide these leading terms: [tex]\(\frac{24x^4}{4x^3} = 6x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(4x^3 + 2x^2\)[/tex] by the term obtained in step 2: [tex]\(6x \cdot (4x^3 + 2x^2) = 24x^4 + 12x^3\)[/tex].
- Subtract this product from the dividend:
[tex]\[
(24x^4 - 24x^3 - 18x^2) - (24x^4 + 12x^3) = -36x^3 - 18x^2.
\][/tex]
4. Repeat the process with the new polynomial:
- The new leading term of the resulting polynomial [tex]\(-36x^3 - 18x^2\)[/tex] is [tex]\(-36x^3\)[/tex].
- Divide the leading terms: [tex]\(\frac{-36x^3}{4x^3} = -9\)[/tex].
- Multiply the entire divisor [tex]\(4x^3 + 2x^2\)[/tex] by the term obtained: [tex]\(-9 \cdot (4x^3 + 2x^2) = -36x^3 - 18x^2\)[/tex].
- Subtract this product from the polynomial:
[tex]\[
(-36x^3 - 18x^2) - (-36x^3 - 18x^2) = 0.
\][/tex]
5. Write the quotient and the remainder:
- The quotient from the division process is [tex]\(6x - 9\)[/tex].
- Since the remainder after the subtraction is [tex]\(0\)[/tex], there is no remainder.
So the final quotient and remainder are:
[tex]\[
\boxed{6x - 9 \quad \text{remainder } 0}
\][/tex]
Therefore, the result of dividing [tex]\(24x^4 - 24x^3 - 18x^2\)[/tex] by [tex]\(4x^3 + 2x^2\)[/tex] is:
[tex]\[
6x - 9
\][/tex] with a remainder of [tex]\(0\)[/tex].
