Chapter 2.3, Problem 88E

To determine

To find:if the given statement is true or false.

Answer to Problem 88E

True

Explanation of Solution

Given:

The statement: “ $\left(2x-1\right)$ is a factor of the polynomial $6x^6+x^5-92x^4+45x^3+184x^2+4x-48$ ”.

Concept Used:

Factor theorem:

"For a polynomial $P\left(x\right)$ , r is a zero of a polynomial if and only if $x-r$ is a factor of $P\left(x\right)$ ”.

By factor theorem, if $\left(2x-1\right)$ is a factor of the polynomial $p\left(x\right)=6x^6+x^5-92x^4+45x^3+184x^2+4x-48$ then $x=\frac{1}{2}$ must be a zero of this polynomial.

That is, it must be true that $p\left(\frac{1}{2}\right)=0$

So, substitute $x=\frac{1}{2}$ in $p\left(x\right)=6x^6+x^5-92x^4+45x^3+184x^2+4x-48$

  $\begin{array}{c}p\left(\frac{1}{2}\right)=6{\left(\frac{1}{2}\right)}^6+{\left(\frac{1}{2}\right)}^5-92{\left(\frac{1}{2}\right)}^4+45{\left(\frac{1}{2}\right)}^3+184{\left(\frac{1}{2}\right)}^2+4\left(\frac{1}{2}\right)-48\\ =\frac{6}{64}+\frac{1}{32}-\frac{92}{16}+\frac{45}{8}+\frac{184}{4}+2-48\\ =0\end{array}$

This implies that $\left(2x-1\right)$ is a factor of the polynomial $6x^6+x^5-92x^4+45x^3+184x^2+4x-48$ .

Conclusion:

The statement is true.