Chapter 4, Problem 15CR

To determine

To calculate: The expression using polynomial long division or synthetic division.

Answer to Problem 15CR

The required value is $x-1+\frac{4x-3}{x^2+2x+1}$ .

Explanation of Solution

Given information:

The expression: $\left(x^3+x^2+3x-4\right)÷\left(x^2+2x+1\right)$

Calculation:

The given expression is $\left(x^3+x^2+3x-4\right)÷\left(x^2+2x+1\right)$ .

Since, the degree of the above polynomial is not 1.

So, we apply polynomial division on the above expression.

  $x^2+2x+1\begin{array}{c}\hfill x-1\\ \hfill \overline{)\begin{array}{l}{x}^3+x^2+3x-4\\ \underset{\_}{x^3+2x^2+x}\\ -x^2+2x-4\\ \underset{\_}{-x^2-2x-1}\\ 4x-3\end{array}}\end{array}$

The quotient of the polynomial is $x-1$ .

Also, the remainder is $4x-3$ .

Thus, $\left(x^3+x^2+3x-4\right)÷\left(x^2+2x+1\right)=x-1+\frac{4x-3}{x^2+2x+1}$

Hence, the required value is $x-1+\frac{4x-3}{x^2+2x+1}$ .