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Question: How do you expand \[{\left( {a - b} \right)^3}\]?...

How do you expand (a−b)3{\left( {a - b} \right)^3}?

Explanation

Solution

Here we have to expand the given expression. Also we use the binomial expansion formula to find the required answer. On further simplification we get the required answer.

Formula used: (x+y)n=xn+nC1xn−1y+nC2xn−2y2+nC3xn−3y3+...+yn{(x + y)^n} = {x^n}{ + ^n}{C_1}{x^{n - 1}}y{ + ^n}{C_2}{x^{n - 2}}{y^2}{ + ^n}{C_3}{x^{n - 3}}{y^3} + ... + {y^n}

Complete step-by-step solution:
We have to find the value of (a−b)3{\left( {a - b} \right)^3}.
We will find the value by binomial expansion formula.
According to the theorem, it is possible to expand the polynomial (x+y)n{(x + y)^n} into a sum involving terms of the form axbyca{x^b}{y^c} where the exponents b and c are nonnegative integers with b+c=nb + c = n, and the coefficient a of each term is a specific positive integer depending on n and b.
We know that,
(x+y)n=xn+nC1xn−1y+nC2xn−2y2+nC3xn−3y3+...+yn{(x + y)^n} = {x^n}{ + ^n}{C_1}{x^{n - 1}}y{ + ^n}{C_2}{x^{n - 2}}{y^2}{ + ^n}{C_3}{x^{n - 3}}{y^3} + ... + {y^n}
For, (x+y)3{\left( {x + y} \right)^3} we have,
⇒(x+y)3=x3+3C1x3−1y+3C2x3−2y2+y3\Rightarrow {\left( {x + y} \right)^3} = {x^3}{ + ^3}{C_1}{x^{3 - 1}}y{ + ^3}{C_2}{x^{3 - 2}}{y^2} + {y^3}
We know that,
3C1=3;3C2=3^3{C_1} = 3{;^3}{C_2} = 3
Using these values and simplifying we get,
⇒(x+y)3=x3+3x2y+3xy2+y3\Rightarrow {\left( {x + y} \right)^3} = {x^3} + 3{x^2}y + 3x{y^2} + {y^3}
Now substitute x=a,y=−bx = a,y = - b we get,
⇒(a−b)3=a3−3a2b+3ab2−b3\Rightarrow {\left( {a - b} \right)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}
Hence, we get
⇒(a−b)3=a3−3a2b+3ab2−b3\Rightarrow {\left( {a - b} \right)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}

Thus, (a−b)3=a3−3a2b+3ab2−b3{\left( {a - b} \right)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3} is the expansion of the given term.

Note: Binomial theorem, statement that for any positive integer n, the nth power of the sum of two numbers a and b may be expressed as the sum of n + 1 terms of the form

n \\\ r \end{array}} \right){a^{n - r}}{b^r}$$ In the sequence of terms, the index r takes on the successive values 0, 1, 2, …, n. The coefficients, called the binomial coefficients, are defined by the formula $$\left( {\begin{array}{*{20}{c}} n \\\ r \end{array}} \right) = \dfrac{{n!}}{{(n - r)!r!}}$$ In which n! (Called n factorial) is the product of the first n natural numbers 1, 2, 3, …, n (and where 0! is defined as equal to 1). The coefficients may also be found in the array often called Pascal’s triangle by finding the $r^{th}$ entry of the $n^{th}$ row (counting begins with a zero in both directions). Each entry in the interior of Pascal’s triangle is the sum of the two entries above it.