Question

How do you find the product ${\left( {2a - b} \right)^3}$?

Answer 1

In the given question and other questions like these to find the product follow the following steps:
Multiply the first term of each binomial together
Multiply the outer terms together
Multiply the inner terms together
Multiply the last term of each expression together
List the four results of FOIL in order
Combine the like terms

Complete Step-By-step solution:
There may be a more efficient and compact way, and someone may explain it, but I’d tend to just brute-force it-
${\left( {2a - b} \right)^3} = \left( {2a - b} \right)\left( {2a - b} \right)\left( {2a - b} \right)$
Ignore the third bracket for now and do ‘FOIL (first, outers, inners, lasts) on the first two brackets:
$\left( {2a - b} \right)\left( {2a - b} \right)\left( {2a - b} \right) = \left( {4{a^2} - 2ab - 2ab + {b^2}} \right)\left( {2a - b} \right)$
Collect like terms:
$\left( {4{a^2} - 4ab + {b^2}} \right)\left( {2a - b} \right)$
Now multiply each term in the left bracket by each term in the right:
$8{a^3} - 4{a^2}b - 8{a^2}b + 4a{b^2} + 2a{b^2} - {b^3}$
Collect like terms again:
$8{a^3} - 12{a^2}b + 6a{b^2} - {b^3}$

Note: In elementary algebra, FOIL is a mnemonic for the standard method of multiplying two binomials – hence the methods may be referred to as the FOIL method. The word FOIL is an acronym for the four terms of the product:
First (first terms of each binomial are multiplied together) Outer (outside terms are multiplied – that is, the first term of the first binomial and the second term of the second) Inner (inside terms are multiplied- second term of the first binomial and first term of the second) Last (last terms of each binomial are multiplied)