Certainly! Let's use synthetic division to rewrite the given fraction [tex]\(\frac{3x^5 - 14x^4 + 14x^3 - 18x^2 - 24x}{x-4}\)[/tex] in the form [tex]\(q(x) + \frac{r(x)}{d(x)}\)[/tex], where [tex]\(d(x)\)[/tex] is the denominator of the original fraction, [tex]\(q(x)\)[/tex] is the quotient, and [tex]\(r(x)\)[/tex] is the remainder.
1. Setup: Write down the coefficients of the dividend polynomial [tex]\(3x^5 - 14x^4 + 14x^3 - 18x^2 - 24x + 0\)[/tex] (with the missing constant term [tex]\(0\)[/tex]) and the root of the divisor [tex]\(x-4=0\)[/tex], which gives us [tex]\(x=4\)[/tex].
Coefficients: [tex]\[3, -14, 14, -18, -24, 0\][/tex]
Divisor root: [tex]\(4\)[/tex]
2. Synthetic Division Table:
```
4 | 3 -14 14 -18 -24 0
| 12 -8 24 24 24
--------------------------------
3 -2 6 6 0 24
```
- Write the first coefficient down (which is the leading coefficient of the polynomial): [tex]\(3\)[/tex].
- Multiply it by the root of the divisor ([tex]\(3 \times 4 = 12\)[/tex]), and place this result under the next coefficient.
- Add the second coefficient [tex]\(-14\)[/tex] and [tex]\(12\)[/tex] giving [tex]\(-2\)[/tex].
- Multiply [tex]\(-2\)[/tex] by [tex]\(4\)[/tex] and write the result [tex]\(-8\)[/tex] under the next coefficient.
- Continue this process of multiplication, addition, and writing down the results until you reach the end.
3. Result:
After completing the synthetic division table, we obtain:
Quotient: [tex]\[3, -2, 6, 6, 0\][/tex]
Remainder: [tex]\(0\)[/tex]
So, the quotient polynomial [tex]\(q(x)\)[/tex] is:
[tex]\[q(x) = 3x^4 - 2x^3 + 6x^2 + 6x\][/tex]
And the remainder [tex]\(r(x)\)[/tex] is:
[tex]\[r(x) = 0\][/tex]
4. Conclusion:
Therefore, the given fraction [tex]\(\frac{3x^5 - 14x^4 + 14x^3 - 18x^2 - 24x}{x-4}\)[/tex] can be written in the form:
[tex]\[
q(x) + \frac{r(x)}{d(x)} = 3x^4 - 2x^3 + 6x^2 + 6x + \frac{0}{x-4}
\][/tex]
5. Final Answer:
[tex]\[
\frac{3x^5 - 14x^4 + 14x^3 - 18x^2 - 24x}{x-4} = 3x^4 - 2x^3 + 6x^2 + 6x
\][/tex]
This fully expanded out includes the quotient polynomial with the remainder simplified.
