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))

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 \(3\) (since \(x - 3 = 0\) gives \(x = 3\)) for synthetic division.<br /><br />Set up the synthetic division:<br /><br />\[<br />\begin{array}{r|rrrrr}<br />3 & 1 & 3 & -18 & 1 & 3 \\<br /> & & 3 & 18 & 0 & 3 \\<br />\hline<br /> & 1 & 6 & 0 & 1 & 6 \\<br />\end{array}<br />\]<br /><br />Here's the step-by-step process:<br /><br />- Bring down the first coefficient (1) directly.<br />- Multiply 3 by the value just written below the line (1), and write the result (3) under the next coefficient (3).<br />- Add the column: \(3 + 3 = 6\).<br />- Multiply 3 by the new value (6), and write the result (18) under the next coefficient (-18).<br />- Add the column: \(-18 + 18 = 0\).<br />- Multiply 3 by the new value (0), and write the result (0) under the next coefficient (1).<br />- Add the column: \(1 + 0 = 1\).<br />- Multiply 3 by the new value (1), and write the result (3) under the next coefficient (3).<br />- Add the column: \(3 + 3 = 6\).<br /><br />The final row gives the coefficients of the quotient polynomial and the remainder:<br /><br />- The quotient polynomial is \(x^3 + 6x^2 + 0x + 1 = x^3 + 6x^2 + 1\).<br />- The remainder is 6.<br /><br />So, the result of the division is:<br /><br />\[<br />x^3 + 6x^2 + 1 + \frac{6}{x - 3}<br />\]<br /><br />Thus, the final answer is:<br /><br />\[<br />x^3 + 6x^2 + 1 + \frac{6}{x - 3}<br />\]