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Question: If \({\text{x - y = - 6}}\) and \({\text{xy = 4}}\), find the value of \({{\text{x}}^3}{\text{ - }}{...

If x - y = - 6{\text{x - y = - 6}} and xy = 4{\text{xy = 4}}, find the value of x3 - y3{{\text{x}}^3}{\text{ - }}{{\text{y}}^3}
a) −288 - 288
b) 288288
c) −28 - 28
d) None of these

Explanation

Solution

In this question we have to find the value of the given equation. For that we are going to solve the problem by using the binomial expansion.
Here binomial expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n{\left( {{\text{x + y}}} \right)^{\text{n}}} into a sum involving terms of the form axbyc{\text{a}}{{\text{x}}^{\text{b}}}{{\text{y}}^{\text{c}}}, where the exponents b{\text{b}} and c{\text{c}} are nonnegative integers with b + c = n{\text{b + c = n}}, and the coefficients a of each term is a specific positive integer depending on n{\text{n}}and b{\text{b}}.
Here we expand the given equation to find the value of it.

Formula used: (x - y)3= x3 - y3 - 3x2y + 3xy2{\left( {{\text{x - y}}} \right)^3} = {\text{ }}{{\text{x}}^3}{\text{ - }}{{\text{y}}^3}{\text{ - 3}}{{\text{x}}^2}{\text{y + 3x}}{{\text{y}}^2}

Complete step-by-step answer:
Given that x - y = - 6{\text{x - y = - 6}} and xy = 4{\text{xy = 4}},
To find the value of equation x3 - y3{{\text{x}}^3}{\text{ - }}{{\text{y}}^3}
Now, consider (x - y)3= x3 - y3 - 3x2y + 3xy2{\left( {{\text{x - y}}} \right)^3} = {\text{ }}{{\text{x}}^3}{\text{ - }}{{\text{y}}^3}{\text{ - 3}}{{\text{x}}^2}{\text{y + 3x}}{{\text{y}}^2}
Substitute the value of terms of x - y = - 6{\text{x - y = - 6}} in this equation,
Then, we have that
(−6)3= x3 - y3 - 3x2y + 3xy2{\left( { - 6} \right)^3} = {\text{ }}{{\text{x}}^3}{\text{ - }}{{\text{y}}^3}{\text{ - 3}}{{\text{x}}^2}{\text{y + 3x}}{{\text{y}}^2}
By substitute the value of terms of (6)3{\left( 6 \right)^3} in the equation,
−216= x3 - y3 - 3x2y + 3xy2- 216 = {\text{ }}{{\text{x}}^3}{\text{ - }}{{\text{y}}^3}{\text{ - 3}}{{\text{x}}^2}{\text{y + 3x}}{{\text{y}}^2}
Taking common terms in this equation,
−216= x3 - y3 - 3yx(x - y)- 216 = {\text{ }}{{\text{x}}^3}{\text{ - }}{{\text{y}}^3}{\text{ - 3yx}}\left( {{\text{x - y}}} \right)
Substitute the value of given equation,
−216= x3 - y3 - 3(4)(−6)- 216 = {\text{ }}{{\text{x}}^3}{\text{ - }}{{\text{y}}^3}{\text{ - 3}}\left( 4 \right)\left( { - 6} \right)
Multiplication of terms in this equation,
x3 - y3 = - 216 + 3(4)(−6){{\text{x}}^3}{\text{ - }}{{\text{y}}^3}{\text{ = - 216 + 3}}\left( 4 \right)\left( { - 6} \right)
By adding the terms in this equation, we get
x3 - y3 = - 288{{\text{x}}^3}{\text{ - }}{{\text{y}}^3}{\text{ = - 288}}

∴\therefore The option A is the correct answer.

Note: A binomial is a polynomial with two variables. It describes the algebraic expansion of the powers. The theorem and its generalizations can be used to prove results and solve problems in combinatorics, algebra, calculus, and many other areas of mathematics.