4)) 2. Use synthetic division to divide 3x^3-5x+10 by x-1 square vour answor

To use synthetic division to divide \(3x^3 - 5x + 10\) by \(x - 1\), we follow these steps:<br /><br />1. Write down the coefficients of the polynomial \(3x^3 - 5x + 10\). Note that we need to include coefficients for all powers of \(x\) from \(x^3\) down to the constant term, even if some coefficients are zero. The polynomial can be written as \(3x^3 + 0x^2 - 5x + 10\), so the coefficients are \(3, 0, -5, 10\).<br /><br />2. Set up the synthetic division. The divisor is \(x - 1\), so we use \(1\) (the root of the divisor) in the synthetic division.<br /><br />\[<br />\begin{array}{r|rrrr}<br />1 & 3 & 0 & -5 & 10 \\<br />\hline<br /> & & & & \\<br />\end{array}<br />\]<br /><br />3. Bring down the first coefficient (3) directly below the line.<br /><br />\[<br />\begin{array}{r|rrrr}<br />1 & 3 & 0 & -5 & 10 \\<br />\hline<br /> & 3 & & & \\<br />\end{array}<br />\]<br /><br />4. Multiply the number just written below the line by the divisor (1) and write the result under the next coefficient.<br /><br />\[<br />\begin{array}{r|rrrr}<br />1 & 3 & 0 & -5 & 10 \\<br />\hline<br /> & 3 & 3 & & \\<br />\end{array}<br />\]<br /><br />5. Add the column of numbers and write the result below the line.<br /><br />\[<br />\begin{array}{r|rrrr}<br />1 & 3 & 0 & -5 & 10 \\<br />\hline<br /> & 3 & 3 & -2 & \\<br />\end{array}<br />\]<br /><br />6. Repeat the process: multiply the last number written below the line (-2) by the divisor (1) and write the result under the next coefficient.<br /><br />\[<br />\begin{array}{r|rrrr}<br />1 & 3 & 0 & -5 & 10 \\<br />\hline<br /> & 3 & 3 & -2 & -2 \\<br />\end{array}<br />\]<br /><br />7. Add the column of numbers and write the result below the line.<br /><br />\[<br />\begin{array}{r|rrrr}<br />1 & 3 & 0 & -5 & 10 \\<br />\hline<br /> & 3 & 3 & -2 & 8 \\<br />\end{array}<br />\]<br /><br />The numbers below the line represent the coefficients of the quotient polynomial and the remainder. The quotient polynomial is \(3x^2 + 3x - 2\) and the remainder is 8.<br /><br />So, the result of dividing \(3x^3 - 5x + 10\) by \(x - 1\) using synthetic division is:<br /><br />\[<br />3x^2 + 3x - 2 + \frac{8}{x - 1}<br />\]