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Expanding a Binomial · Physics Playground

Exercises Sequences, counting and probability

Example 2 from Algebra and Trigonometry, 13.6 Binomial Theorem

Expanding a Binomial

Write in expanded form.

ⓐ ${(x + y)}^{5}$
ⓑ ${(3 x - y)}^{4}$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. ⓐSubstitute $n = 5$ into the formula. Evaluate the $k = 0$ through $k = 5$ terms. Simplify.
  $\begin{array}{l} {(x + y)}^{5} & = (\begin{matrix} 5 \\ 0 \end{matrix}) x^{5} y^{0} + (\begin{matrix} 5 \\ 1 \end{matrix}) x^{4} y^{1} + (\begin{matrix} 5 \\ 2 \end{matrix}) x^{3} y^{2} + (\begin{matrix} 5 \\ 3 \end{matrix}) x^{2} y^{3} + (\begin{matrix} 5 \\ 4 \end{matrix}) x^{1} y^{4} + (\begin{matrix} 5 \\ 5 \end{matrix}) x^{0} y^{5} \\ {(x + y)}^{5} & = x^{5} + 5 x^{4} y + 10 x^{3} y^{2} + 10 x^{2} y^{3} + 5 x y^{4} + y^{5} \end{array}$
2. ⓑSubstitute $n = 4$ into the formula. Evaluate the $k = 0$ through $k = 4$ terms. Notice that $3 x$ is in the place that was occupied by $x$ and that $- y$ is in the place that was occupied by $y .$ So we substitute them. Simplify.
  $\begin{array}{l} {(3 x - y)}^{4} & = (\begin{matrix} 4 \\ 0 \end{matrix}) {(3 x)}^{4} {(- y)}^{0} + (\begin{matrix} 4 \\ 1 \end{matrix}) {(3 x)}^{3} {(- y)}^{1} + (\begin{matrix} 4 \\ 2 \end{matrix}) {(3 x)}^{2} {(- y)}^{2} + (\begin{matrix} 4 \\ 3 \end{matrix}) {(3 x)}^{1} {(- y)}^{3} + (\begin{matrix} 4 \\ 4 \end{matrix}) {(3 x)}^{0} {(- y)}^{4} \\ {(3 x - y)}^{4} & = 81 x^{4} - 108 x^{3} y + 54 x^{2} y^{2} - 12 x y^{3} + y^{4} \end{array}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Write in expanded form.

ⓐ ${(x - y)}^{5}$
ⓑ ${(2 x + 5 y)}^{3}$

How did it go?

PreviousFinding Binomial Coefficients Next Writing a Given Term of a Binomial Expansion

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 13.6, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.