se synthetic division to simplify \( \frac{9 x^{3}-25 x^{2}+6}{x-3} \) Vrite your answer in the form \( \mathrm{q}(\mathrm{x})+\frac{\mathrm{r}}{\mathrm{d}(\mathrm{x})} \), where \( \mathrm{q}(\mathrm{x}) \) is a polynomial, r is an nteger, and \( \mathrm{d}(\mathrm{x}) \) is a linear polynomial. Simplify any fractions.
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To simplify \( \frac{9 x^{3}-25 x^{2}+6}{x-3} \) using synthetic division, we start with the coefficients of the polynomial \( 9, -25, 0, 6 \). 1. Write down the 3 (the root from \( x - 3 \)) on the left and the coefficients on the right: ``` 3 | 9 -25 0 6 | ``` 2. Bring down the 9: ``` 3 | 9 -25 0 6 | | 9 ``` 3. Multiply 3 by 9, which gives 27, and add it to -25: ``` 3 | 9 -25 0 6 | 27 | 9 2 ``` 4. Now multiply 3 by 2, which gives 6, and add it to 0: ``` 3 | 9 -25 0 6 | 27 6 | 9 2 6 ``` 5. Finally, multiply 3 by 6, which gives 18, and add it to 6: ``` 3 | 9 -25 0 6 | 27 6 | 9 2 24 ``` The final row gives us the coefficients of the quotient and the remainder. Thus, we have: - \( q(x) = 9x^2 + 2x + 6\) - \( r = 24\) Putting it all together, we write our answer as: \[ q(x) + \frac{r}{d(x)} = 9x^2 + 2x + 6 + \frac{24}{x-3} \]
