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Answer :
To solve the division of the polynomial [tex]\(4x^5 - 39x^3 + 33x - 16\)[/tex] by [tex]\(x - 3\)[/tex] using synthetic division, follow these steps:
1. Set up the synthetic division:
- First, notice that the divisor is [tex]\(x - 3\)[/tex], which means the root to use in synthetic division is [tex]\(3\)[/tex].
- The dividend polynomial is [tex]\(4x^5 + 0x^4 - 39x^3 + 0x^2 + 33x - 16\)[/tex]. Notice we include zero coefficients for the missing [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] terms.
2. Write the coefficients of the dividend: [4, 0, -39, 0, 33, -16].
3. Perform synthetic division:
- Bring down the leading coefficient [tex]\(4\)[/tex] to the bottom row.
- Multiply [tex]\(4\)[/tex] by [tex]\(3\)[/tex] (the root from [tex]\(x - 3\)[/tex]) and write the result under the next coefficient, which is [tex]\(0\)[/tex].
- Add [tex]\(0 + 12 = 12\)[/tex], and write [tex]\(12\)[/tex] on the bottom row.
- Multiply [tex]\(12\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(36\)[/tex], and add this to [tex]\(-39\)[/tex] to get [tex]\(-3\)[/tex].
- Multiply [tex]\(-3\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(-9\)[/tex], and add this to [tex]\(0\)[/tex] to get [tex]\(-9\)[/tex].
- Multiply [tex]\(-9\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(-27\)[/tex], and add this to [tex]\(33\)[/tex] to get [tex]\(6\)[/tex].
- Finally, multiply [tex]\(6\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(18\)[/tex], and add this to [tex]\(-16\)[/tex] to get a remainder of 2.
4. Write the result:
The coefficients at the bottom row represent the resulting polynomial: [tex]\(4x^4 + 12x^3 - 3x^2 - 9x + 6\)[/tex], and the remainder is [tex]\(2\)[/tex].
5. Write the final answer in the requested form:
[tex]\[
\text{The quotient is } q(x) = 4x^4 + 12x^3 - 3x^2 - 9x + 6
\][/tex]
[tex]\[
\text{The remainder is } r = 2
\][/tex]
[tex]\[
\text{So the division results in: } q(x) + \frac{r}{d(x)} = 4x^4 + 12x^3 - 3x^2 - 9x + 6 + \frac{2}{x-3}
\][/tex]
This is the complete solution using synthetic division to express [tex]\((4x^5 - 39x^3 + 33x - 16) \div (x-3)\)[/tex] in the form of [tex]\(q(x) + \frac{r}{d(x)}\)[/tex].
1. Set up the synthetic division:
- First, notice that the divisor is [tex]\(x - 3\)[/tex], which means the root to use in synthetic division is [tex]\(3\)[/tex].
- The dividend polynomial is [tex]\(4x^5 + 0x^4 - 39x^3 + 0x^2 + 33x - 16\)[/tex]. Notice we include zero coefficients for the missing [tex]\(x^4\)[/tex] and [tex]\(x^2\)[/tex] terms.
2. Write the coefficients of the dividend: [4, 0, -39, 0, 33, -16].
3. Perform synthetic division:
- Bring down the leading coefficient [tex]\(4\)[/tex] to the bottom row.
- Multiply [tex]\(4\)[/tex] by [tex]\(3\)[/tex] (the root from [tex]\(x - 3\)[/tex]) and write the result under the next coefficient, which is [tex]\(0\)[/tex].
- Add [tex]\(0 + 12 = 12\)[/tex], and write [tex]\(12\)[/tex] on the bottom row.
- Multiply [tex]\(12\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(36\)[/tex], and add this to [tex]\(-39\)[/tex] to get [tex]\(-3\)[/tex].
- Multiply [tex]\(-3\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(-9\)[/tex], and add this to [tex]\(0\)[/tex] to get [tex]\(-9\)[/tex].
- Multiply [tex]\(-9\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(-27\)[/tex], and add this to [tex]\(33\)[/tex] to get [tex]\(6\)[/tex].
- Finally, multiply [tex]\(6\)[/tex] by [tex]\(3\)[/tex] to get [tex]\(18\)[/tex], and add this to [tex]\(-16\)[/tex] to get a remainder of 2.
4. Write the result:
The coefficients at the bottom row represent the resulting polynomial: [tex]\(4x^4 + 12x^3 - 3x^2 - 9x + 6\)[/tex], and the remainder is [tex]\(2\)[/tex].
5. Write the final answer in the requested form:
[tex]\[
\text{The quotient is } q(x) = 4x^4 + 12x^3 - 3x^2 - 9x + 6
\][/tex]
[tex]\[
\text{The remainder is } r = 2
\][/tex]
[tex]\[
\text{So the division results in: } q(x) + \frac{r}{d(x)} = 4x^4 + 12x^3 - 3x^2 - 9x + 6 + \frac{2}{x-3}
\][/tex]
This is the complete solution using synthetic division to express [tex]\((4x^5 - 39x^3 + 33x - 16) \div (x-3)\)[/tex] in the form of [tex]\(q(x) + \frac{r}{d(x)}\)[/tex].
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