Chapter 12: Q.12.69TI (page 1209)

Use the Binomial Theorem to expand ${(x + y)}^{5}$

Short Answer

Expert verified

The expanded form of ${(x + y)}^{5}$ is,

$x^{5} + 5 x^{4} y + 10 x^{3} y^{2} + 10 x^{2} y^{3} + 5 x y^{4} + y^{5}$

Step by step solution

Given information

Use the Binomial Theorem to expand ${(x + y)}^{5}$

on using the binomial theorem

${(x + y)}^{5} = (\begin{matrix} 5 \\ 0 \end{matrix}) x^{5} + (\begin{matrix} 5 \\ 1 \end{matrix}) x^{4} y + (\begin{matrix} 5 \\ 2 \end{matrix}) x^{3} y^{2} + (\begin{matrix} 5 \\ 3 \end{matrix}) x^{2} y^{3} + (\begin{matrix} 5 \\ 4 \end{matrix}) x y^{4} + (\begin{matrix} 5 \\ 5 \end{matrix}) y^{5} a s w e k n o w, (\begin{matrix} n \\ 0 \end{matrix}) = (\begin{matrix} n \\ n \end{matrix}) = 1 a n d, (\begin{matrix} n \\ 1 \end{matrix}) = n {(x + y)}^{5} = x^{5} + 5 x^{4} y + \frac{5!}{2! (5 - 2)!} x^{3} y^{2} + \frac{5!}{3! (5 - 3)!} x^{2} y^{3} + \frac{5!}{4! (5 - 4)!} x y^{4} + y^{5} = x^{5} + 5 x^{4} y + \frac{5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 3 \times 2 \times 1} x^{3} y^{2} + \frac{5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 2 \times 1} x^{2} y^{3} + \frac{5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1 \times 1} x y^{4} + y^{5} = x^{5} + 5 x^{4} y + 10 x^{3} y^{2} + 10 x^{2} y^{3} + 5 x y^{4} + y^{5}$

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Use the Binomial Theorem to expand ${(x + y)}^{5}$

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Use the Binomial Theorem to expandx+y5

Short Answer

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on using the binomial theorem

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Most popular questions from this chapter

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Use the Binomial Theorem to expand ${(x + y)}^{5}$