Use synthetic division to find the result when 3x^4-x^3+5x^2+x-8 is divided by x-1 Answer Attemptiout of 3 square

To use synthetic division to divide the polynomial \(3x^4 - x^3 + 5x^2 + x - 8\) by \(x - 1\), we follow these steps:<br /><br />1. Write down the coefficients of the polynomial: \(3, -1, 5, 1, -8\).<br />2. Write the value of \(c\) from \(x - c\) on the left. Here, \(c = 1\).<br /><br />Set up the synthetic division:<br /><br />\[<br />\begin{array}{r|rrrrr}<br />1 & 3 & -1 & 5 & 1 & -8 \\<br /> & & 3 & 2 & 7 & 8 \\<br />\hline<br /> & 3 & 2 & 7 & 8 & 0 \\<br />\end{array}<br />\]<br /><br />Here's the step-by-step process:<br /><br />- Bring down the first coefficient (3) directly.<br />- Multiply the value just written below the line (3) by \(c\) (1), and write the result under the next coefficient: \(3 \times 1 = 3\).<br />- Add the column: \(-1 + 3 = 2\).<br />- Repeat the process: Multiply the new value (2) by \(c\) (1), and write the result under the next coefficient: \(2 \times 1 = 2\).<br />- Add the column: \(5 + 2 = 7\).<br />- Repeat the process: Multiply the new value (7) by \(c\) (1), and write the result under the next coefficient: \(7 \times 1 = 7\).<br />- Add the column: \(1 + 7 = 8\).<br />- Repeat the process: Multiply the new value (8) by \(c\) (1), and write the result under the next coefficient: \(8 \times 1 = 8\).<br />- Add the column: \(-8 + 8 = 0\).<br /><br />The remainder is 0, which confirms that \(x - 1\) is a factor of the polynomial.<br /><br />The quotient is the polynomial formed by the coefficients below the line, excluding the remainder:<br /><br />\[<br />3x^3 + 2x^2 + 7x + 8<br />\]<br /><br />So, the result of dividing \(3x^4 - x^3 + 5x^2 + x - 8\) by \(x - 1\) using synthetic division is:<br /><br />\[<br />3x^3 + 2x^2 + 7x + 8<br />\]