Chapter B, Problem 1E

To determine

To fill: The blank provided in the statement, “The factored form of $2x^3+x^2-x+10$ is $\left(x+2\right)\left(2x^2-3x+5\right)$. The remainder theorem assures us that when $2x^3+x^2-x+10$ is divided by $\left(x+2\right)$ the remainder is ________.”

Answer to Problem 1E

Solution:

The complete statement is,“The factored form of $2x^3+x^2-x+10$ is $\left(x+2\right)\left(2x^2-3x+5\right)$. The remainder theorem assures us that when $2x^3+x^2-x+10$ is divided by $\left(x+2\right)$ the remainder is 0.”

Explanation of Solution

Remainder theorem states that, when the polynomial $P\left(x\right)$ is divided by $x-k$ then the remainder is equal to $P\left(k\right)$.

When $P\left(x\right)=2x^3+x^2-x+10$ is divided by $x-\left(-2\right)$, then remainder should be 0.

Using synthetic division as,

$-2\overline{)\begin{array}{c}\begin{array}{cccc}2& 1& -1& 10\end{array}\\ \underset{\_}{\begin{array}{cccc}& -4& 6& -10\end{array}}\\ \begin{array}{cccc}2& -3& 5& 0\end{array}\end{array}}$

The remainder is 0.

$P\left(-2\right)=0$

Hence, the complete statement is,“The factored form of $2x^3+x^2-x+10$ is $\left(x+2\right)\left(2x^2-3x+5\right)$. The remainder theorem assures us that when $2x^3+x^2-x+10$ is divided by $\left(x+2\right)$ the remainder is 0.”