Question

If $p\left(x\right)=3x^2-2x+5\text{and}r\left(x\right)=x^3+x+1$ then find $r\left(x+1\right)$.

Short answer

The value of $r\left(x+1\right)=x^3+3x^2+4x+3$.

Step-by-step solution

Step 1. Write down the given information.

The given functions are $p\left(x\right)=3x^2-2x+5\text{and}r\left(x\right)=x^3+x+1$.

Step 2. Calculation.

Evaluating $r\left(x+1\right)$ by plugging $x=x+1$ in given function $r\left(x\right)=x^3+x+1$.

$\begin{array}{c}r\left(x\right)=x^3+x+\mathrm{1....}\left(\text{Given}\right)\\ r\left(x+1\right)={\left(x+1\right)}^3+\left(x+1\right)+1\\ =x^3+1+3x^2+3x+x+1+1\\ =x^3+3x^2+4x+3\end{array}$

Step 3. Conclusion.

The value of $r\left(x+1\right)=x^3+3x^2+4x+3$.