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Chapter 2: Problem 97

Find the value of \( k \) such that \( x - 4 \) is a factor of $ x^3 - kx^2 + 2kx - 8.

Short Answer

Expert verified
The value of \( k \) that makes \( x - 4 \) a factor of \( x^3 - kx^2 + 2kx - 8 \) is \( k = 1 \)

Step by step solution

01

Use the factor theorem

Since \( x - 4 \) is a factor of the polynomial \( x^3 - kx^2 + 2kx - 8 \), substitute \( x = 4 \) in the equation to get \( 4^3 - 4k \cdot 4^2 + 2k \cdot 4 - 8 = 0 \)
02

Simplify the equation

After the substitution, simplify the equation to get \( 64 - 64k + 8k - 8 = 0 \) or \( -56k + 56 = 0 \)
03

Solve for k

Solve for \( k \) by adding \( 56k \) to both sides and then dividing both sides by \( 56 \) to get \( k = 1 \)

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In Exercises 13 - 30, solve the inequality and graph the solution on the real number line. \( x^2 + x < 6 \)

In Exercises 85 - 87, determine whether the statement is true or false. Justify your answer. The graph of a rational function can never cross one of its asymptotes.

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