Question Use synthetic division to find the result when x^4+3x^3-18x^2+x+3 is divided by x-3 If there is a remainder , express the result in the form q(x)+(r(x))/(b(x)) Answer Anemptiout of a square

To use synthetic division to divide \(x^4 + 3x^3 - 18x^2 + x + 3\) by \(x - 3\), we follow these steps:<br /><br />1. Write down the coefficients of the polynomial: \(1, 3, -18, 1, 3\).<br />2. Use the root of the divisor \(x - 3\), which is \(3\).<br /><br />Set up the synthetic division:<br /><br />```<br /> 3 | 1 3 -18 1 3<br /> | 3 18 0 -3<br /> -------------------<br /> 1 6 0 1 0<br />```<br /><br />Here's the step-by-step process:<br /><br />- Bring down the first coefficient (1).<br />- Multiply 3 by the value just written below the line (1), and write the result under the next coefficient (3).<br />- Add the values in the second column: \(3 + 3 = 6\).<br />- Multiply 3 by the new value (6), and write the result under the next coefficient (-18).<br />- Add the values in the third column: \(-18 + 18 = 0\).<br />- Multiply 3 by the new value (0), and write the result under the next coefficient (1).<br />- Add the values in the fourth column: \(1 + 0 = 1\).<br />- Multiply 3 by the new value (1), and write the result under the next coefficient (3).<br />- Add the values in the fifth column: \(3 + 3 = 6\).<br /><br />The final row gives the coefficients of the quotient polynomial and the remainder. The quotient polynomial is \(x^3 + 6x^2 + 0x + 1\) and the remainder is 0.<br /><br />So, the result of the division is:<br /><br />\[ x^3 + 6x^2 + 1 \]<br /><br />Since the remainder is 0, there is no need to express the result in the form \(q(x) + \frac{r(x)}{b(x)}\). The final answer is simply:<br /><br />\[ x^3 + 6x^2 + 1 \]