Question

How do you multiply \[{\left( {3x - 2} \right)^2}\]?

Answer 1

Hint: Here, we will find the product of the given Binomials. We will multiply the binomials by using the horizontal method or FOIL method to find the product of their Binomials. Thus, the product of the given Binomials is the required answer. A binomial expression is defined as an algebraic expression having two terms and these terms must be unlike.

Complete step by step solution:
We are given an algebraic expression \[{\left( {3x - 2} \right)^2}\].
Now, we will multiply the given algebraic expression twice since the power is 2.
\[ \Rightarrow {\left( {3x - 2} \right)^2} = \left( {3x - 2} \right)\left( {3x - 2} \right)\]
We will find the product of the given binomials by using the horizontal method or FOIL method of multiplying one term of a binomial with each other term in another binomial.
By multiplying each term in one binomial with each other term in the other binomial, we get
\[ \Rightarrow {\left( {3x - 2} \right)^2} = 3x\left( {3x - 2} \right) - 2\left( {3x - 2} \right)\]
By multiplying the terms, we get
\[ \Rightarrow {\left( {3x - 2} \right)^2} = 9{x^2} - 6x - 6x + 4\]
By adding the like terms, we get
\[ \Rightarrow {\left( {3x - 2} \right)^2} = 9{x^2} - 12x + 4\]

Therefore, the product of \[\left( {3x - 2} \right) \times \left( {3x - 2} \right)\] is \[9{x^2} - 12x + 4\].

Additional Information:
The FOIL method is a method of multiplying the binomials by multiplying the first terms, then the outer terms, then the inner terms, and at last the last terms. It is possible to combine like terms. Thus, the product of two binomials is a trinomial.

Note:
We know that we can also find the product by using an algebraic identity. The square of the difference between two numbers is given by an algebraic identity \[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\].
By substituting \[a = 3x\] and \[b = 2\] , we get
\[ \Rightarrow {\left( {3x - 2} \right)^2} = {\left( {3x} \right)^2} + {\left( 2 \right)^2} - 2\left( {3x} \right)\left( 2 \right)\].
By simplifying the equation, we get
\[ \Rightarrow {\left( {3x - 2} \right)^2} = 9{x^2} + 4 - 12x\]
Thus, we get \[{\left( {3x - 2} \right)^2} = 9{x^2} - 12x + 4\].