Question

How do you multiply \[\left( 3n+2 \right)\left( n+3 \right)\]?

Answer 1

Hint: We will use FOIL technique to solve this problem. First we have to multiply both terms according to FOIL rule. After that we had to combine the like terms and had to simplify the equation obtained to arrive at the solution.

Complete step by step answer:
Before solving the problem, we have to know the FOIL rule.
FOIL rule: In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials hence the method may be referred to as the FOIL method.
The word FOIL is an acronym for the four terms of the product:
F stands for First that means first terms of each binomial are multiplied together
O stands for Outer that means outside terms are multiplied that is the first term of the first binomial and the second term of the second
I stands for Inner that means inside terms are multiplied that is the second term of the first binomial and first term of the second
L stands for Last that means last terms of each binomial are multiplied
So the given equation is \[\left( 3n+2 \right)\left( n+3 \right)\]
According to the FOIL principle we have to multiply both binomials.
First we have to multiply first terms of both the binomials. It will look like
\[\Rightarrow 3n\times n\]
\[\Rightarrow 3{{n}^{2}}\]
Now we have to multiply the first term of the first binomial and the second term of the second binomial. Then the equation will be like
\[\Rightarrow 3{{n}^{2}}+3n\times 3\]
\[\Rightarrow 3{{n}^{2}}+9n\]
Now we have to multiply the second term of the first binomial and the first term of the second binomial. Then the equation will be like
\[\Rightarrow 3{{n}^{2}}+9n+2\times n\]
\[\Rightarrow 3{{n}^{2}}+9n+2n\]
Now the last step of multiplying the second term in both binomials
\[\Rightarrow 3{{n}^{2}}+9n+2n+2\times 3\]
\[\Rightarrow 3{{n}^{2}}+9n+2n+6\]
Now we have to combine like terms in the equation. Then the equation will look like
\[\Rightarrow 3{{n}^{2}}+n\left( 9+2 \right)+6\]
\[3{{n}^{2}}+11n+6\]
So by multiplying the given binomials we will get \[3{{n}^{2}}+11n+6\].
We get \[\left( 3n+2 \right)\left( n+3 \right)\Rightarrow 3{{n}^{2}}+11n+6\] .

Note:
We can also solve it by the direct multiplication method we do. This is the basic question solved in polynomials so knowledge about polynomials will be helpful. The equation we obtained is a quadratic equation so we can get values of n by solving the equation.