Question

How do you multiply \[\left( {2{p^3} + 5{p^2} - 4} \right)\left( {3p + 1} \right)\]?

Answer 1

Hint:
We will first use the fact that (a + b + c) (x + y) = a (x + y) + b (x + y) + c (x + y) and then we will use the distributive property which states that u (v + w) = uv + uw.

Complete step by step solution:
We are given that we are required to find the value of \[\left( {2{p^3} + 5{p^2} - 4} \right)\left( {3p + 1} \right)\].
We know that we have a formula given by the following expression:-
$ \Rightarrow $(a + b + c) (x + y) = a (x + y) + b (x + y) + c (x + y)
Replacing a by $2{p^3}$, b by $5{p^2}$, c by – 4, x by 3p and y by 1, we will then obtain the following expression with us:-
\[ \Rightarrow \left( {2{p^3} + 5{p^2} - 4} \right)\left( {3p + 1} \right) = 2{p^3}\left( {3p + 1} \right) + 5{p^2}\left( {3p + 1} \right) - 4\left( {3p + 1} \right)\] ………………(1)
Now, we will use distributive property in \[2{p^3}\left( {3p + 1} \right)\], we will then obtain the following expression with us:-
\[ \Rightarrow 2{p^3}\left( {3p + 1} \right) = 2{p^3} \times 3p + 2{p^3} \times 1\]
Simplifying the right hand side of the above expression, we will then obtain the following expression with us:-
\[ \Rightarrow 2{p^3}\left( {3p + 1} \right) = 6{p^4} + 2{p^3}\] …………..(2)
Now, we will use distributive property in \[5{p^2}\left( {3p + 1} \right)\], we will then obtain the following expression with us:-
\[ \Rightarrow 5{p^2}\left( {3p + 1} \right) = 5{p^2} \times 3p + 5{p^2} \times 1\]
Simplifying the right hand side of the above expression, we will then obtain the following expression with us:-
\[ \Rightarrow 5{p^2}\left( {3p + 1} \right) = 15{p^3} + 5{p^2}\] …………..(3)
Now, we will use distributive property in \[ - 4\left( {3p + 1} \right)\], we will then obtain the following expression with us:-
\[ \Rightarrow - 4\left( {3p + 1} \right) = - 4 \times 3p + ( - 4) \times 1\]
Simplifying the right hand side of the above expression, we will then obtain the following expression with us:-
\[ \Rightarrow - 4\left( {3p + 1} \right) = - 12p - 4\] …………..(4)
Putting the equation number 2, 3 and 4 in equation number 1, we will then obtain the following equation:-
\[ \Rightarrow \left( {2{p^3} + 5{p^2} - 4} \right)\left( {3p + 1} \right) = 6{p^4} + 2{p^3} + 15{p^3} + 5{p^2} - 12p - 4\]
Simplifying the right hand side of the above equation by clubbing the like terms, we will then obtain the following expression:-
\[ \Rightarrow \left( {2{p^3} + 5{p^2} - 4} \right)\left( {3p + 1} \right) = 6{p^4} + 17{p^3} + 5{p^2} - 12p - 4\]
Thus, we have the required answer.

Note:
The students must remember the fact given by the following expression:-
$ \Rightarrow $(a + b + c) (x + y) = a (x + y) + b (x + y) + c (x + y)
The students must also note that we have used the distributive property as we mentioned in the solution. It is given as follows:-
For any numbers a, b and c, we have the following expression:-
$ \Rightarrow $ a (b + c) = ab + ac