Question

If \[a - b = 4\] and \[ab = 21\], find the value of \[{a^3} - {b^3}\]

Answer 1

Hint: In this question, we have to find the value of \[{a^3} - {b^3}\]. Here, we have given \[a\] minus \[b\] is \[4\] and multiplication of \[a\] and \[b\] is \[21\]. For getting the value of \[{a^3} - {b^3}\], we re-arrange the identity of \[{\left( {a - b} \right)^3}\]. Firstly, expand the identity, after that re-arrange the unknown terms and take them to one side and known terms to the other side. The, put the value of known terms; after putting the value, do the calculation (addition and subtraction). Then we will get the unknown terms.

Complete step-by-step answer:
Step 1: Given \[a - b = 4\] and \[ab = 21\]
We have to find the value of \[{a^3} - {b^3}\]
We know the identity of \[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right)\]; on the right-hand side of the identity, the first two terms are \[{a^3} - {b^3}\]. To find the value of \[{a^3} - {b^3}\], take the other terms \[3ab\left( {a - b} \right)\] on the left-hand side.
Therefore, \[{\left( {a - b} \right)^3} + 3ab\left( {a - b} \right) = {a^3} - {b^3}\]………(A)
On the left-hand side, the first term has \[\left( {a - b} \right)\] and the second term has both \[ab\] and \[a - b\]. We have the value of \[\left( {a - b} \right)\] and \[ab\]. So, put the value of \[\left( {a - b} \right)\] and \[ab\] in equation (A), we get:
\[{\left( 4 \right)^3} + 3 \times 21 \times 4 = {a^3} - {b^3}\]
We know that the cube of \[4\] is \[64\] and multiplication of \[3,21{\text{ and }}4\] is \[252\].
Therefore, \[64 + 252 = {a^3} - {b^3}\]
After addition of \[64{\text{ and }}252\], we get \[316\].
That is, \[316 = {a^3} - {b^3}\]
Hence the value of \[{a^3} - {b^3}\] is \[316\].

Note: A polynomial of one term is called a monomial. A polynomial of two terms is called a binomial. A polynomial of three terms is called a trinomial. A polynomial of degree three is called a cubic polynomial. Every linear polynomial in one variable has a unique zero, a non-zero constant polynomial has no zero and every real number is a zero of the zero polynomial. A polynomial \[p\left( x \right)\] in one variable \[x\] is an algebraic expression in \[x\] of the form \[p\left( x \right) = {a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ..... + {a_1}x + {a_0}x\].