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Chapter 2: Problem 96

Find the constant \(c\) such that the denominator will divide evenly into the numerator. \(\frac{x^{5}-2 x^{2}+x+c}{x+2}\)

Short Answer

Expert verified
The value of \(c\) is determined by performing polynomial division on the function and equating the constant term that is left after conducting polynomial division to zero.

Step by step solution

01

Perform Polynomial Division

We start by performing the division of the given function. Just like in simple arithmetic, where you would divide numbers, there is a similar process to divide polynomials. We start by dividing the first term in the numerator by the first term in the denominator. This gives us \((x^5 / x) = x^4\). Multiply the divisor (\(x+2\)) by \(x^4\) and subtract the product from the polynomial under the division. This leads to a new polynomial \(x^4 - 2x^2 + x + c\).
02

Repeat the Division Process

We now repeat the process on the new polynomial. This gives us \((x^4 / x) = x^3\). Multiply the divisor (\(x+2\)) by \(x^3\) and subtract the product from the polynomial under the division. Then, again, we subtract, getting a new polynomial. We continue this process till we get the constant term left.
03

Find Value of C

After performing the polynomial division and getting the constant term left, equate the constant term to zero and solve for \(c\). This is because we want the denominator to be a factor of the numerator, which is equivalent to the function having no remainder when the numerator is divided by the denominator.

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