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Chapter 2: Problem 53

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k,\) and demonstrate that \(f(k)=r\). \(f(x)=-4 x^{3}+6 x^{2}+12 x+4, \quad k=1-\sqrt{3}\)

Short Answer

Expert verified
The solution to the exercise is the function \(f(x)\) in the form \((x-k)q(x)+r\) and the verification of the statement \(f(k) = r\). The specific functions for \(q(x)\) and \(r\) will be determined in Step 2 and substituted into the equation in Step 3. The check for the validity of the equation will be performed in Step 4.

Step by step solution

01

Identify the function and the given value of k

We're given the function \(f(x)=-4 x^{3}+6 x^{2}+12 x+4\) and the value \(k=1-\sqrt{3}\)
02

Divide the function by \((x-k)\)

Using polynomial division or synthetic division, divide \(f(x)\) by \((x-(1-\sqrt3))\). Simplifying \(x-(1-\sqrt3)\) gives \(x-1+\sqrt3\). The result of this division is \(q(x)\), the quotient, and \(r\), the remainder.
03

Write \(f(x)\) in the form \((x-k)q(x)+r\)

The result of the division provides us with the functions \(q(x)\) and \(r\), which we can now substitute back into the equation \(f(x) = (x-k)q(x) + r\) to form the correct equation.
04

Validate the result

Substitute \(k\) into \(f(x)\) and compute the value. Check that this is equal to the remainder \(r\) calculated. If both values match, it verifies that the rewritten function is correct and the exercise is done correctly.

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Most popular questions from this chapter

Use a graphing utility to graph the equation. Use the graph to approximate the values of \(x\) that satisfy each inequality. \(y=\frac{1}{2} x^{2}-2 x+1 \quad\) (a) \(y \leq 0 \quad\) (b) \(y \geq 7\)

Determine whether each value of \(x\) is a solution of the inequality. Inequality. \(\begin{array}{lll}\frac{x+2}{x-4} \geq 3 & \text { (a) } x=5 & \text { (b) } x=4 \\ & \text { (c) } x=-\frac{9}{2} & \text { (d) } x=\frac{9}{2}\end{array}\)

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