Use synthetic division to find the result when 3x^3-10x^2-21x-20 is divided by x-5

To use synthetic division to divide \(3x^3 - 10x^2 - 21x - 20\) by \(x - 5\), we follow these steps:<br /><br />1. Write down the coefficients of the polynomial: \(3, -10, -21, -20\).<br />2. Write the zero of the divisor \(x - 5\), which is \(5\), to the left.<br /><br />Set up the synthetic division:<br /><br />\[<br />\begin{array}{r|rrrr}<br />5 & 3 & -10 & -21 & -20 \\<br /> & & 15 & 25 & 20 \\<br />\hline<br /> & 3 & 5 & 4 & 0 \\<br />\end{array}<br />\]<br /><br />Here's the step-by-step process:<br /><br />- Bring down the first coefficient \(3\).<br />- Multiply \(3\) by \(5\) to get \(15\).<br />- Add \(-10\) and \(15\) to get \(5\).<br />- Multiply \(5\) by \(5\) to get \(25\).<br />- Add \(-21\) and \(25\) to get \(4\).<br />- Multiply \(4\) by \(5\) to get \(20\).<br />- Add \(-20\) and \(20\) to get \(0\).<br /><br />The final row gives the coefficients of the quotient polynomial and the remainder. Since the remainder is \(0\), the quotient polynomial is:<br /><br />\[ 3x^2 + 5x + 4 \]<br /><br />So, the result of dividing \(3x^3 - 10x^2 - 21x - 20\) by \(x - 5\) using synthetic division is:<br /><br />\[ 3x^2 + 5x + 4 \]