Question

How do you use the binomial formula to expand \[{{\left( 2x+3 \right)}^{3}}?\]

Answer 1

Note:
We are asked to expand \[{{\left( 2x+3 \right)}^{3}}\] using the binomial formula. To answer this we will learn what binomial means and what binomial formulas are. Once we know that we will focus on the part that which formula will suit and we consider a = 2x and b = 3 and solve our problem. We will use the formula \[{{\left( a+b \right)}^{3}}={{a}^{3}}+3{{a}^{2}}b+3{{b}^{2}}a+{{b}^{3}}.\]

Complete step by step answer:
We have to expand \[{{\left( 2x+3 \right)}^{3}},\] we start by learning about the binomial. A pair consisting of two terms are known as binomial. Here b is usually used to denote 2. So, the binomial is a group of two terms. The binomial formula is those formulas that are applied on the pair of two terms. For different powers, we have different formulas. For example, if we have power 2, so then we have \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\] and if the terms are subtracting, then \[{{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\] and if the power is 3 then we have \[{{\left( a+b \right)}^{3}}={{a}^{3}}+3{{a}^{2}}b+3{{b}^{2}}a+{{b}^{3}}.\] Now we will check what our problem is. We have \[{{\left( 2x+3 \right)}^{3}}\] and here the power as we see is 3. So we will use \[{{\left( a+b \right)}^{3}}.\] Now to solve our problem we will use the formula \[{{\left( a+b \right)}^{3}},\] we let a = 2x and b = 3. So we will get, \[{{\left( 2x+3 \right)}^{3}}={{\left( 2x \right)}^{3}}+3\times {{\left( 2x \right)}^{2}}\left( 3 \right)+3\left( 2x \right){{3}^{2}}+{{3}^{3}}\]
Now, as \[{{\left( 2x \right)}^{3}}={{2}^{3}}.{{x}^{3}}\] and \[{{\left( 2x \right)}^{2}}={{2}^{2}}{{x}^{2}}.\] So, we get,
\[\Rightarrow {{\left( 2x+3 \right)}^{3}}={{2}^{3}}{{x}^{3}}+3\times 3\times {{2}^{2}}{{x}^{2}}+3\times {{3}^{2}}\times 2x+{{3}^{3}}\]
On simplifying, we get,
\[\Rightarrow {{\left( 2x+3 \right)}^{3}}=8{{x}^{3}}+36{{x}^{2}}+54x+27\]
So, we get the expanded form of \[{{\left( 2x+3 \right)}^{3}}\] as \[8{{x}^{3}}+36{{x}^{2}}+54x+27.\]

Note:
 We need to have good knowledge of the formulas like \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}\] or \[{{\left( a-b \right)}^{2}}={{a}^{2}}-{{b}^{2}}\] will lead us into incorrect solutions. We also need to understand that \[{{x}^{3}}=x\times x\times x\] and we should not do it like \[{{x}^{3}},3x\] or \[{{x}^{2}}=2x\] as they also are incorrect options. While adding, always remember that like terms can only be added and unlike terms are not allowed to add or subtract.