Question

use synthetic division to simplify \\(\frac{x^4 - 10x^3 - 21x^2 + 10}{x + 1}\\).
write your answer in the form \\(q(x) + \frac{r}{d(x)}\\), where \\(q(x)\\) is a polynomial, \\(r\\) is an integer, and \\(d(x)\\) is a linear polynomial. simplify any fractions.

Explanation:

Step1: Identify the root for division

For the divisor \(x + 1\), the root is \(c=-1\) (since \(x+1 = x-(-1)\)). The coefficients of the dividend \(x^{4}-10x^{3}-21x^{2}+0x + 10\) (we add the missing \(x\) term with coefficient 0) are \(1, - 10, -21, 0, 10\).

Step2: Perform synthetic division

Set up the synthetic division:

$$
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$$

• Bring down the first coefficient: \(1\).
• Multiply by \(-1\): \(1\times(-1)=-1\), add to the next coefficient: \(-10+(-1)=-11\).
• Multiply \(-11\) by \(-1\): \(-11\times(-1) = 11\), add to the next coefficient: \(-21 + 11=-10\).
• Multiply \(-10\) by \(-1\): \(-10\times(-1)=10\), add to the next coefficient: \(0 + 10 = 10\).
• Multiply \(10\) by \(-1\): \(10\times(-1)=-10\), add to the last coefficient: \(10+(-10)=0\).

The quotient polynomial \(q(x)\) has coefficients \(1, -11, -10, 10\), so \(q(x)=x^{3}-11x^{2}-10x + 10\), the remainder \(r = 0\), and the divisor \(d(x)=x + 1\).

Answer:

\(x^{3}-11x^{2}-10x + 10+\frac{0}{x + 1}\) (or simply \(x^{3}-11x^{2}-10x + 10\) since the remainder term is \(0\))