The Remainder and Factor Theorem Score: 7.3/10 Answered: 8/10 Question 10 Given that f(x)=-5x^3-7x^2+6x+4 use synthetic division and the Remainder Theorem to find f(-8) f(-8)=square

To find \( f(-8) \) using synthetic division and the Remainder Theorem, we follow these steps:<br /><br />1. **Set up the synthetic division**: Write down the coefficients of \( f(x) \) and the value \( x = -8 \).<br /><br /> Coefficients: \(-5, -7, 6, 4\)<br /><br /> Value: \(-8\)<br /><br />2. **Perform the synthetic division**:<br /><br /> ```<br /> -8 | -5 -7 6 4<br /> | 80 -576 4608<br /> -------------------<br /> -5 73 -570 4612<br /> ```<br /><br /> Here's the step-by-step process:<br /> - Bring down the first coefficient: \(-5\).<br /> - Multiply \(-8\) by \(-5\) to get \(40\), then add to the next coefficient: \(-7 + 40 = 33\).<br /> - Multiply \(-8\) by \(33\) to get \(-264\), then add to the next coefficient: \(6 - 264 = -258\).<br /> - Multiply \(-8\) by \(-258\) to get \(2056\), then add to the next coefficient: \(4 + 2056 = 2060\).<br /><br />3. **The remainder is the last value**: The remainder when \( f(x) \) is divided by \( x + 8 \) is \( 2060 \).<br /><br />4. **Apply the Remainder Theorem**: According to the Remainder Theorem, \( f(-8) \) is equal to the remainder obtained from the synthetic division.<br /><br />Therefore, \( f(-8) = 2060 \).<br /><br />So, the answer is:<br />\[ f(-8) = 2060 \]