Problem 51 Write the function in the form \... [FREE SOLUTION]

Chapter 2: Problem 51

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

Short Answer

Expert verified

After performing polynomial division and substituting \(k = \sqrt{2}\) into \(f(k)\), if \(f(\sqrt{2}) = r\) (remainder from the division), then the polynomial has been correctly divided and can be expressed in the form \(f(x) = (x-\sqrt{2}) q(x) + r\).

Step by step solution

Perform Polynomial Division

Begin the process by performing polynomial division. Divide \(f(x)\) by \((x-\sqrt{2})\) using long division or synthetic division to get \(q(x)\) and a remainder \(r\).

Check for Correctness

The function will be in the form \(f(x) = (x-k) q(x) + r\) when \((x-k)\) is completely factored out. Now, the equation can be formally written. In this case, \(f(x) = (x-\sqrt{2}) q(x) + r\). Make sure that the \(q(x)\) and \(r\) you got in the first step makes this equation true.

Substituting Value of \(k\)

Substitute \(k = \sqrt{2}\) into \(f(k)\) and compare it with the remainder \(r\) from the polynomial division. If \(f(\sqrt{2})\) equals to the remainder \(r\), then the polynomial division is correct, and the equation \(f(x) = (x-k) q(x) + r\) is properly factored.

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91影视

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

Short Answer

Step by step solution

Perform Polynomial Division

Check for Correctness

Substituting Value of \(k\)

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