Chapter 12: Q 28. (page 836)

Expand ${(a x - b y)}^{4}$ using Binomial Theorem

Short Answer

Expert verified

The value of ${(a x - b y)}^{4} = a^{4} x^{4} - 4 a^{3} b x^{3} y + 6 a^{2} b^{2} x^{2} y^{2} - 4 a b^{3} x y^{3} + b^{4} y^{4}$

Step by step solution

Step 1. Defination

Let $x$ and $a$ be real number for any positive integer $n$ , we have ${(x + a)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) x^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a x^{n - 1} + \dots \dots + (\begin{matrix} n \\ j \end{matrix}) a^{j} x^{n - j} + \dots \dots + (\begin{matrix} n \\ n \end{matrix}) a^{n} = \sum_{j = 0}^{n} (\begin{matrix} n \\ j \end{matrix}) x^{n - j} a^{j}$

Step 2. Explanation

$\Rightarrow {(a x - b y)}^{4} = (\begin{matrix} 4 \\ 0 \end{matrix}) {(- b y)}^{0} {(a x)}^{4} + (\begin{matrix} 4 \\ 1 \end{matrix}) {(- b y)}^{1} {(a x)}^{3} + (\begin{matrix} 4 \\ 2 \end{matrix}) {(- b y)}^{2} {(a x)}^{2} + (\begin{matrix} 4 \\ 3 \end{matrix}) {(- b y)}^{3} {(a x)}^{1} + (\begin{matrix} 4 \\ 4 \end{matrix}) {(- b y)}^{4} {(a x)}^{0}$

$\Rightarrow {(a x - b y)}^{4} = a^{4} x^{4} - 4 a^{3} b x^{3} y + 6 a^{2} b^{2} x^{2} y^{2} - 4 a b^{3} x y^{3} + b^{4} y^{4}$

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Expand ${(a x - b y)}^{4}$ using Binomial Theorem

Short Answer

Step by step solution

Step 1. Defination

Step 2. Explanation

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Expand ax-by4using Binomial Theorem

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Expand ${(a x - b y)}^{4}$ using Binomial Theorem