Questions: (5x^4 - 2x^2 + 2x + 9) / (x^3 - x + 6)

(5x^4 - 2x^2 + 2x + 9) / (x^3 - x + 6)
Transcript text: \[ \frac{5 x^{4}-2 x^{2}+2 x+9}{x^{3}-x+6} \]
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Solution

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Solution Steps

To solve the problem of finding the remainder when dividing \(5x^4 - 2x^2 + 2x + 9\) by \(x^3 - x + 6\) using long division, we need to perform polynomial long division. This involves dividing the terms of the dividend by the leading term of the divisor and subtracting the result from the dividend iteratively until the degree of the remainder is less than the degree of the divisor.

Step 1: Define the Polynomials

We start with the dividend \(5x^4 - 2x^2 + 2x + 9\) and the divisor \(x^3 - x + 6\).

Step 2: Perform Polynomial Long Division

We divide the leading term of the dividend \(5x^4\) by the leading term of the divisor \(x^3\), which gives us \(5x\). We then multiply the entire divisor by \(5x\) and subtract this from the dividend.

\[ 5x \cdot (x^3 - x + 6) = 5x^4 - 5x^2 + 30x \]

Subtracting this from the dividend:

\[ (5x^4 - 2x^2 + 2x + 9) - (5x^4 - 5x^2 + 30x) = 3x^2 - 28x + 9 \]

Step 3: Continue the Division

Next, we take the new polynomial \(3x^2 - 28x + 9\) and divide its leading term \(3x^2\) by the leading term of the divisor \(x^3\). Since the degree of the new polynomial is less than that of the divisor, we stop here. The remainder is \(3x^2 - 28x + 9\).

Final Answer

The remainder when dividing \(5x^4 - 2x^2 + 2x + 9\) by \(x^3 - x + 6\) is

\[ \boxed{3x^2 - 28x + 9} \]

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