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Chapter 2: Problem 97

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

Short Answer

Expert verified
The value of \(k\) that makes \(x-4\) a factor of the polynomial \(x^{3}-k x^{2}+2 k x-8\) is \(k = 7\).

Step by step solution

01

Identify the value for substitution

Given that \(x-4\) is a factor, according to the Factor Theorem, when \(x = 4\), the polynomial \(x^{3}-k x^{2}+2 k x-8\) should equal 0.
02

Substitute the value of \(x\)

Substitute \(x = 4\) into the polynomial: \(4^{3}-k*4^{2}+2 k*4-8 = 0\). This simplifies to \(64-16k+8k-8 = 0\).
03

Solve the Equation

Combining like terms, the equation becomes: \(56 - 8k = 0\). Solving for \(k\) gives \(k = \frac{56}{8} = 7\).

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. $$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=-1$$

Use the given zero to find all the zeros of the function. Function \(f(x)=x^{3}-x^{2}+4 x-4\) Zero \(2 i\)

Use synthetic division to verify the upper and lower bounds of the real zeros of \(f\) \(f(x)=2 x^{4}-8 x+3\) (a) Upper: \(x=3\) (b) Lower: \(x=-4\)

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-3 x^{3}-x^{2}-12 x-20\) (Hint: One factor is \(\left.x^{2}+4 .\right)\)

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(s)=2 s^{3}-5 s^{2}+12 s-5$$
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