Question

use synthetic division to find the result when 2x⁴ − 7x² − 4x + 2 is divided by x + 1. if there is a remainder, express the result in the form q(x) + (r(x))/(b(x)). answer attempt 1 out of 4

Explanation:

Step1: Identify the root for synthetic division

For the divisor \(x + 1\), the root \(c\) is \(-1\) (since \(x+1 = 0\) gives \(x=-1\)). The coefficients of the dividend \(2x^{4}-7x^{2}-4x + 2\) are \(2\) (for \(x^{4}\)), \(0\) (for \(x^{3}\), since there is no \(x^{3}\) term), \(-7\) (for \(x^{2}\)), \(-4\) (for \(x\)), and \(2\) (constant term).

Step2: Set up synthetic division

We set up the synthetic division as follows:

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• Bring down the leading coefficient \(2\).
• Multiply \(2\) by \(-1\) to get \(-2\), add to the next coefficient \(0\): \(0+(-2)=-2\).
• Multiply \(-2\) by \(-1\) to get \(2\), add to the next coefficient \(-7\): \(-7 + 2=-5\).
• Multiply \(-5\) by \(-1\) to get \(5\), add to the next coefficient \(-4\): \(-4+5 = 1\).
• Multiply \(1\) by \(-1\) to get \(-1\), add to the last coefficient \(2\): \(2+(-1)=1\).

Step3: Write the quotient and remainder

The coefficients of the quotient \(q(x)\) are \(2\) (for \(x^{3}\)), \(-2\) (for \(x^{2}\)), \(-5\) (for \(x\)), and \(1\) (constant term), so \(q(x)=2x^{3}-2x^{2}-5x + 1\). The remainder \(r(x)\) is \(1\), and the divisor \(b(x)=x + 1\).

So the result of the division is \(2x^{3}-2x^{2}-5x + 1+\frac{1}{x + 1}\).

Answer:

\(2x^{3}-2x^{2}-5x + 1+\frac{1}{x + 1}\)