Question

If $p\left(x\right)=3x^2-2x+5\text{and}r\left(x\right)=x^3+x+1$ then find $3\left[p\left(x^2-1\right)\right]+4p\left(x\right)$.

Short answer

The value of $3\left[p\left(x^2-1\right)\right]+\text{4}\left[p\left(x\right)\right]=9x^4-12x^2-8x+50$.

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 $3\left[p\left(x^2-1\right)\right]+4p\left(x\right)$by plugging $x=x^2-1$in given function $p\left(x\right)=3x^2-2x+5$and multiply it by 3 and then add $4\left[p\left(x\right)\right]$.

Therefore,

role="math" localid="1648124984708" $\begin{array}{c}p\left(x\right)=3x^2-2x+5\mathrm{....}\left(\text{Given}\right)\\ 3\left[p\left(x\right)\right]=3\left[3x^2-2x+5\right]\\ 3\left[p\left(x^2-1\right)\right]=3\left[3{\left(x^2-1\right)}^2-2\left(x^2-1\right)+5\right]\\ =3\left[3x^4-8x^2+10\right]\\ =9x^4-24x^2+30\mathrm{....}\left(1\right)\\ \text{Now,4}\left[p\left(x\right)\right]=4\left[3x^2-2x+5\right]\\ =12x^2-8x+20\mathrm{....}\left(2\right)\end{array}$

Adding (1) and (2) gives,

$\begin{array}{c}3\left[p\left(x^2-1\right)\right]+\text{4}\left[p\left(x\right)\right]=9x^4-24x^2+30+12x^2-8x+20\\ =9x^4-12x^2-8x+50\end{array}$

Step 3. Conclusion.

The value of $3\left[p\left(x^2-1\right)\right]+\text{4}\left[p\left(x\right)\right]=9x^4-12x^2-8x+50$.