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Chapter 2: Problem 47

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k,\) and demonstrate that \(f(k)=r\). \(f(x)=x^{3}-x^{2}-14 x+11, \quad k=4\)

Short Answer

Expert verified
By dividing the given polynomial by the linear factor and upon substitution we get the remaining value -3 which is our remainder. Hence the polynomial in the required form is \(f(x)=(x-4)(x^{2}+3x+2) -3\) and when substituted with the given \(k\) value, we find that \(f(k)=r\), which verifies the polynomial division theorem.

Step by step solution

01

Polynomial Division

First, you divide the polynomial \(f(x) = x^{3}-x^{2}-14 x+11\) by the linear factor \(x-k\), in which \(k=4\). Hence, we divide by \(x-4\). Using synthetic or long division methods for polynomials, we find the quotient \(q(x) =x^{2}+3x+2\) and remainder \(r=-3\). Therefore the polynomial can be written as \(f(x)=(x-4)(x^{2}+3x+2) -3\).
02

Substitution

Now, Substitute \(k\) in the function \(f(x)\), in place of \(x\). Hence substitute \(k=4\) in the function \(f(4)=(4-4)(4^{2}+3*4+2) -3\).
03

Comparison

Evaluate the function for \(k=4\), we get \(f(4)=-3\). This matches with our remainder we got in step 1. Hence it's proved that \(f(k)=r\).

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