To divide the polynomial [tex]\( 28x^5 + 29x^4 + 5x^3 + 86x^2 + 56x + 53 \)[/tex] by the linear polynomial [tex]\(-4x - 7\)[/tex] using synthetic division, follow these steps:
1. Identify the root of the divisor: The divisor is [tex]\(-4x - 7\)[/tex]. To find its root, we set [tex]\(-4x - 7 = 0\)[/tex], which gives [tex]\(x = -\frac{7}{-4} = \frac{7}{4}\)[/tex].
2. Set up the synthetic division: We use the coefficients of the polynomial [tex]\(28x^5 + 29x^4 + 5x^3 + 86x^2 + 56x + 53\)[/tex], which are [tex]\(28, 29, 5, 86, 56, 53\)[/tex], and the root [tex]\(\frac{7}{4}\)[/tex].
3. Perform the synthetic division:
- Write down the coefficients: [tex]\(28, 29, 5, 86, 56, 53\)[/tex].
- Bring down the first coefficient [tex]\(28\)[/tex].
- Multiply [tex]\(28\)[/tex] by [tex]\(\frac{7}{4}\)[/tex]: [tex]\(28 \cdot \frac{7}{4} = 49\)[/tex].
- Add this product to the next coefficient: [tex]\(29 + 49 = 78.0\)[/tex].
Repeat these steps:
- Multiply [tex]\(78.0\)[/tex] by [tex]\(\frac{7}{4}\)[/tex]: [tex]\(78.0 \cdot \frac{7}{4} = 136.5\)[/tex].
- Add this product to the next coefficient: [tex]\(5 + 136.5 = 141.5\)[/tex].
- Multiply [tex]\(141.5\)[/tex] by [tex]\(\frac{7}{4}\)[/tex]: [tex]\(141.5 \cdot \frac{7}{4} = 247.125\)[/tex].
- Add this product to the next coefficient: [tex]\(86 + 247.125 = 333.625\)[/tex].
- Multiply [tex]\(333.625\)[/tex] by [tex]\(\frac{7}{4}\)[/tex]: [tex]\(333.625 \cdot \frac{7}{4} = 583.21875\)[/tex].
- Add this product to the next coefficient: [tex]\(56 + 583.21875 = 639.84375\)[/tex].
- Multiply [tex]\(639.84375\)[/tex] by [tex]\(\frac{7}{4}\)[/tex]: [tex]\(639.84375 \cdot \frac{7}{4} = 1116.7265625\)[/tex].
- Add this product to the last coefficient: [tex]\(53 + 1116.7265625 = 1169.7265625\)[/tex].
After completing the steps, the quotient coefficients are:
[tex]\( [28, 78.0, 141.5, 333.625, 639.84375] \)[/tex].
The remainder is:
[tex]\(1172.7265625\)[/tex].
Hence, the quotient [tex]\(Q(x)\)[/tex] is [tex]\( 28x^4 + 78.0x^3 + 141.5x^2 + 333.625x + 639.84375 \)[/tex] and the remainder [tex]\(R\)[/tex] is [tex]\( 1172.7265625 \)[/tex].
