Question

use synthetic division to find the result when $4x^4 + 11x^3 + 17x^2 + x - 9$ is divided by $x + 1$.

Explanation:

Step1: Identify root of divisor

For \(x + 1 = 0\), the root is \(x=-1\). The coefficients of the dividend \(4x^{4}+11x^{3}+17x^{2}+x - 9\) are \(4, 11, 17, 1, -9\).

Step2: Set up synthetic division

Write the root \(-1\) on the left and the coefficients in a row:
\[

$$\begin{array}{r|rrrrr} -1 & 4 & 11 & 17 & 1 & -9 \\ & & -4 & -7 & -10 & 9 \\ \hline & 4 & 7 & 10 & -9 & 0 \\ \end{array}$$

\]

• Bring down the first coefficient \(4\).
• Multiply \(4\) by \(-1\) to get \(-4\), add to \(11\): \(11+(-4)=7\).
• Multiply \(7\) by \(-1\) to get \(-7\), add to \(17\): \(17+(-7)=10\).
• Multiply \(10\) by \(-1\) to get \(-10\), add to \(1\): \(1+(-10)=-9\).
• Multiply \(-9\) by \(-1\) to get \(9\), add to \(-9\): \(-9 + 9=0\) (remainder is \(0\)).

Step3: Write the quotient

The coefficients of the quotient (from the bottom row, excluding the last which is remainder) correspond to a polynomial of degree one less than the dividend. So the quotient is \(4x^{3}+7x^{2}+10x - 9\) (since remainder is \(0\), the division is exact).

Answer:

The result of the division is \(4x^{3}+7x^{2}+10x - 9\) (with a remainder of \(0\)).