Chapter 12: Q. 12.69 (page 1209)

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

Short Answer

Expert verified

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

Step 1. Given information.

The given binomial is ${(x + y)}^{5}$ .

Step 2. Use the binomial theorem.

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

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use the binomial theorem.

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Use the Binomial Theorem to expand x+y5.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Use the binomial theorem.

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

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