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Chapter 2: Problem 74

Find \(k\) so that \(4 x+3\) is a factor of $$20 x^{3}+23 x^{2}-10 x+k$$

Short Answer

Expert verified
The value for \(k\) that makes \(4x+3\) a factor of the cubic polynomial is \(k = \frac{19}{2}\).

Step by step solution

01

Identify the root corresponding to the factor

We have \(4x+3\) with roots \(x=-\frac{3}{4}\). This value of \(x\) will be used according to the factor theorem: if \(4x+3\) is a factor of the polynomial, then the polynomial will equal zero when \(x = -\frac{3}{4}\).
02

Substitute the root into the cubic polynomial and equate to zero

We substitute \(x= -\frac{3}{4}\) into the polynomial \(20x^3+23x^2-10x+k\), yielding \(-\frac{45}{8}-\frac{135}{32}+\frac{15}{4}+k=0\).
03

Simplify and solve for \(k\)

Next we simplify and solve for \(k\), \(k =\frac{45}{8}+\frac{135}{32}-\frac{15}{4}= \frac{19}{2}\).

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