Chapter 12: Q. 12.72. (page 1210)

Use the Binomial Theorem to expand ${(y - 1)}^{6}$ .

Short Answer

Expert verified

The expansion of the given binomial is, $y^{6} - 6 y^{5} + 15 y^{4} - 20 y^{3} + 15 y^{2} - 6 y + 1$

Step by step solution

Step 1. Given information.

The given expression is, ${(y - 1)}^{6}$

Step 2. Substitute the values and simplify.

Substitute in the values into the binomial expansion,

$a = y b = - 1 n = 6$

Now

${(y - 1)}^{6} = (\begin{matrix} 6 \\ 0 \end{matrix}) y^{6} + (\begin{matrix} 6 \\ 1 \end{matrix}) y^{6 - 1} (- 1) + (\begin{matrix} 6 \\ 2 \end{matrix}) y^{6 - 2} {(- 1)}^{2} + (\begin{matrix} 6 \\ 3 \end{matrix}) y^{6 - 3} {(- 1)}^{3} + (\begin{matrix} 6 \\ 4 \end{matrix}) y^{6 - 4} {(- 1)}^{4} + (\begin{matrix} 6 \\ 5 \end{matrix}) y^{6 - 5} {(- 1)}^{5} + (\begin{matrix} 6 \\ 6 \end{matrix}) {(- 1)}^{6}$

Simplify the exponents.

${(y - 1)}^{6} = (\begin{matrix} 6 \\ 0 \end{matrix}) y^{6} - (\begin{matrix} 6 \\ 1 \end{matrix}) y^{5} + (\begin{matrix} 6 \\ 2 \end{matrix}) y^{4} - (\begin{matrix} 6 \\ 3 \end{matrix}) y^{3} + (\begin{matrix} 6 \\ 4 \end{matrix}) y^{2} - (\begin{matrix} 6 \\ 5 \end{matrix}) y + (\begin{matrix} 6 \\ 6 \end{matrix})$

Evaluate the coefficients. Remember,

$(\begin{matrix} n \\ 1 \end{matrix}) = n, (\begin{matrix} n \\ n \end{matrix}) = 1, (\begin{matrix} n \\ 0 \end{matrix}) = 1$

Simplify the exponents,

${(y - 1)}^{6} = 1 y^{6} - 6 y^{5} + \frac{6!}{2! 4!} y^{4} - \frac{6!}{3! 3!} y^{3} + \frac{6!}{4! 2!} y^{2} - \frac{6!}{5! 1!} y + 1$

Step 3. Conclusion.

Therefore, the expansion of the given expression is ${(y - 1)}^{6} = y^{6} - 6 y^{5} + 15 y^{4} - 20 y^{3} + 15 y^{2} - 6 y + 1$

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Use the Binomial Theorem to expand ${(y - 1)}^{6}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitute the values and simplify.

Step 3. Conclusion.

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Step by step solution

Step 1. Given information.

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Step 3. Conclusion.

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Use the Binomial Theorem to expand ${(y - 1)}^{6}$ .