Use synthetic division to divide. (3x^2-7)/(x-2) (3x^2-7)/(x-2)= square

To use synthetic division to divide \(\frac{3x^2 - 7}{x - 2}\), we first need to set up the synthetic division. The divisor is \(x - we use \(2\) as the number for synthetic division.<br /><br />Here are the steps:<br /><br />1. Write down the coefficients of the dividend \(3x^2 - 7\). Since there is no \(x\) term, we include a coefficient of \(0\):<br /> \[<br /> 3 \quad 0 \quad -7<br /> \]<br /><br />2. Write \(2\) to the left and draw a line:<br /> \[<br /> \begin{array}{r|rrr}<br /> 2 & 3 & 0 & -7 \\<br /> & & & \\<br /> \hline<br /> & 3 & & \\<br /> \end{array}<br /> \]<br /><br />3. Bring down the first coefficient (3):<br /> \[<br /> \begin{array}{r|rr 2 & 3 & 0 & -7 \\<br /> & & & \\<br /> \hline<br /> & 3 & & \\<br /> \end{array}<br /> \]<br /><br />4. Multiply \(2\) by the value just written below the line (3), and write the result under the next coefficient:<br /> \[<br /> \begin{array}{r|rrr}<br /> 2 & 3 & 0 & -7 \\<br /> & & 6 & \\<br /> \hline<br /> & 3 & 6 & \\<br /> \end{array}<br /> \]<br /><br />5. Add the column of numbers:<br /> \[<br /> \begin{array}{r|rrr}<br /> 2 & 3 & 0 & -7 \\<br /> &6 & 12 \\<br /> \hline<br /> & 3 & 6 & 5 \\<br /> \end{array}<br /> \]<br /><br />The bottom row gives us the coefficients of the quotient polynomial and the remainder. The quotient is \(3x + 6\) and the remainder is \(5\).<br /><br />So, the result of the division is:<br />\[<br />\frac{3x^2 - 7}{x - 2} = 3x + 6 + \frac{5}{x - 2}<br />\]<br /><br />Thus, the final answer is:<br />\[<br />\boxed{3x + 6 + \frac{5}{x - 2}}<br />\]