Answered

Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.

Use synthetic division to rewrite the following fraction 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.

[tex]\[ \frac{3x^5 - 14x^4 + 14x^3 - 18x^2 - 24x}{x-4} \][/tex]

Sagot :

GinnyAnswer
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.
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.