Chapter 4: Q. 88 (page 210)

Find k such that $f (x) = x^{4} - k x^{3} + k x^{2} + 1$ has the factor $x + 2$ .

Short Answer

Expert verified

The required value of k is $- \frac{17}{12}$ .

Step by step solution

Step 1. Concept Used

The factor theorem states that iff is a polynomial function, then $x - c$ is a factor of f if and only if $f (c) = 0$ .

Step 2. Form an equation

As it is given that $x + 2$ is a factor of $f (x) = x^{4} - k x^{3} + k x^{2} + 1$ .

So by the factor theorem we have $f (- 2) = 0$ .

$\begin{array}{rcl} f (- 2) & = & 0 \\ {(- 2)}^{4} - k {(- 2)}^{3} + k {(- 2)}^{2} + 1 & = & 0 \\ 16 + 8 k + 4 k + 1 & = & 0 \\ 17 + 12 k & = & 0 \end{array}$

Step 3. Solve the equation

In the equation subtract $17$ from both sides

$\begin{array}{rcl} 17 + 12 k - 17 & = & 0 - 17 \\ 12 k & = & - 17 \end{array}$

Now divide both sides by $12$ and solve for k.

$\begin{array}{rcl} \frac{12 k}{12} & = & \frac{- 17}{12} \\ k & = & - \frac{17}{12} \end{array}$

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Find k such that $f (x) = x^{4} - k x^{3} + k x^{2} + 1$ has the factor $x + 2$ .

Short Answer

Step by step solution

Step 1. Concept Used

Step 2. Form an equation

Step 3. Solve the equation

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Find k such that f(x)=x4-kx3+kx2+1has the factorx+2.

Short Answer

Step by step solution

Step 1. Concept Used

Step 2. Form an equation

Step 3. Solve the equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

Find k such that $f (x) = x^{4} - k x^{3} + k x^{2} + 1$ has the factor $x + 2$ .