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Answer :
Sure! Let's go through the process of synthetic division step-by-step to divide the polynomial [tex]\(x^4 - 81\)[/tex] by [tex]\(x + 3\)[/tex].
### Step-by-Step Process:
1. Identify the Dividend and Divisor:
- The dividend is the polynomial [tex]\(x^4 - 81\)[/tex].
- The divisor is [tex]\(x + 3\)[/tex].
2. Convert the Divisor:
- For synthetic division, use the root of the divisor set to zero. So, [tex]\(x + 3 = 0\)[/tex] gives [tex]\(x = -3\)[/tex].
- We'll use [tex]\(-3\)[/tex] in our synthetic division.
3. Set Up the Coefficients:
- Write the coefficients of the dividend. For [tex]\(x^4 - 81\)[/tex], there are missing terms, so insert zeros: [tex]\(1x^4 + 0x^3 + 0x^2 + 0x - 81\)[/tex]. Thus, the coefficients are [tex]\( [1, 0, 0, 0, -81] \)[/tex].
4. Perform Synthetic Division:
- Step 1: Bring down the leading coefficient:
- Start with the first coefficient (1). This becomes the leading term of your quotient.
- Write down: [tex]\(1\)[/tex].
- Step 2: Multiply and Add:
- Multiply the number just written (1) by [tex]\(-3\)[/tex] (the divisor root), resulting in [tex]\(-3\)[/tex].
- Add [tex]\(-3\)[/tex] to the next coefficient (0), yielding [tex]\(-3\)[/tex].
- Write down: [tex]\(-3\)[/tex].
- Repeat:
- Multiply [tex]\(-3\)[/tex] (the result) by [tex]\(-3\)[/tex], resulting in [tex]\(9\)[/tex].
- Add [tex]\(9\)[/tex] to the next coefficient (0), yielding [tex]\(9\)[/tex].
- Write down: [tex]\(9\)[/tex].
- Multiply [tex]\(9\)[/tex] by [tex]\(-3\)[/tex], resulting in [tex]\(-27\)[/tex].
- Add [tex]\(-27\)[/tex] to the next coefficient (0), yielding [tex]\(-27\)[/tex].
- Write down: [tex]\(-27\)[/tex].
- Multiply [tex]\(-27\)[/tex] by [tex]\(-3\)[/tex], resulting in [tex]\(81\)[/tex].
- Add [tex]\(81\)[/tex] to the last coefficient (-81), yielding [tex]\(0\)[/tex].
- Write down the remainder: [tex]\(0\)[/tex].
5. Final Result:
- The quotient is given by the coefficients you've written down, which correspond to one degree less than the original polynomial. This yields [tex]\(x^3 - 3x^2 + 9x - 27\)[/tex].
- The remainder is [tex]\(0\)[/tex], indicating that [tex]\(x + 3\)[/tex] divides [tex]\(x^4 - 81\)[/tex] evenly.
So, the division of [tex]\(x^4 - 81\)[/tex] by [tex]\(x + 3\)[/tex] using synthetic division gives the quotient:
[tex]\[ x^3 - 3x^2 + 9x - 27 \][/tex]
And since the remainder is zero, the polynomial division is exact, confirming that [tex]\(x + 3\)[/tex] is indeed a factor of [tex]\(x^4 - 81\)[/tex].
### Step-by-Step Process:
1. Identify the Dividend and Divisor:
- The dividend is the polynomial [tex]\(x^4 - 81\)[/tex].
- The divisor is [tex]\(x + 3\)[/tex].
2. Convert the Divisor:
- For synthetic division, use the root of the divisor set to zero. So, [tex]\(x + 3 = 0\)[/tex] gives [tex]\(x = -3\)[/tex].
- We'll use [tex]\(-3\)[/tex] in our synthetic division.
3. Set Up the Coefficients:
- Write the coefficients of the dividend. For [tex]\(x^4 - 81\)[/tex], there are missing terms, so insert zeros: [tex]\(1x^4 + 0x^3 + 0x^2 + 0x - 81\)[/tex]. Thus, the coefficients are [tex]\( [1, 0, 0, 0, -81] \)[/tex].
4. Perform Synthetic Division:
- Step 1: Bring down the leading coefficient:
- Start with the first coefficient (1). This becomes the leading term of your quotient.
- Write down: [tex]\(1\)[/tex].
- Step 2: Multiply and Add:
- Multiply the number just written (1) by [tex]\(-3\)[/tex] (the divisor root), resulting in [tex]\(-3\)[/tex].
- Add [tex]\(-3\)[/tex] to the next coefficient (0), yielding [tex]\(-3\)[/tex].
- Write down: [tex]\(-3\)[/tex].
- Repeat:
- Multiply [tex]\(-3\)[/tex] (the result) by [tex]\(-3\)[/tex], resulting in [tex]\(9\)[/tex].
- Add [tex]\(9\)[/tex] to the next coefficient (0), yielding [tex]\(9\)[/tex].
- Write down: [tex]\(9\)[/tex].
- Multiply [tex]\(9\)[/tex] by [tex]\(-3\)[/tex], resulting in [tex]\(-27\)[/tex].
- Add [tex]\(-27\)[/tex] to the next coefficient (0), yielding [tex]\(-27\)[/tex].
- Write down: [tex]\(-27\)[/tex].
- Multiply [tex]\(-27\)[/tex] by [tex]\(-3\)[/tex], resulting in [tex]\(81\)[/tex].
- Add [tex]\(81\)[/tex] to the last coefficient (-81), yielding [tex]\(0\)[/tex].
- Write down the remainder: [tex]\(0\)[/tex].
5. Final Result:
- The quotient is given by the coefficients you've written down, which correspond to one degree less than the original polynomial. This yields [tex]\(x^3 - 3x^2 + 9x - 27\)[/tex].
- The remainder is [tex]\(0\)[/tex], indicating that [tex]\(x + 3\)[/tex] divides [tex]\(x^4 - 81\)[/tex] evenly.
So, the division of [tex]\(x^4 - 81\)[/tex] by [tex]\(x + 3\)[/tex] using synthetic division gives the quotient:
[tex]\[ x^3 - 3x^2 + 9x - 27 \][/tex]
And since the remainder is zero, the polynomial division is exact, confirming that [tex]\(x + 3\)[/tex] is indeed a factor of [tex]\(x^4 - 81\)[/tex].
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