Chapter 8.2, Problem 46HP

To determine

To determine the value of $p$ from the given expression.

Answer to Problem 46HP

  $p=3$

Explanation of Solution

Given:

  $3x^p(4x^{2p+3}+2x^{3p-2})=12x^{12}+6x^{10}$

Calculation:

Consider the given expression,

  $\Rightarrow 3x^p(4x^{2p+3}+2x^{3p-2})=12x^{12}+6x^{10}$

Now, multiplying the common part on the left hand side to make the base of both sides equal,

  $\Rightarrow 3x^p\times 4x^{2p+3}+3x^p\times 2x^{3p-2}=12x^{12}+6x^{10}$

  $\begin{array}{l}\Rightarrow 12x^{p+(2p+3)}+6x^{p+(3p-2)}=12x^{12}+6x^{10}\\ \Rightarrow 12x^{(3p+3)}+6x^{(4p-2)}=12x^{12}+6x^{10}\end{array}$

Comparing coefficients of  $x^{12}$ and  $x^{10}$ on both sides as the base value of the powers are same then:

  $\begin{array}{l}{x}^{12}:3p+3=12\Rightarrow p=3\\ x^{10}:4p-2=10\Rightarrow p=3\end{array}$

[ $\because$ solving for the value of variable $p$ ]

Conclusion:

Therefore, the value of $p$ for the given expression is $p=3$ .