Chapter 12: Q. 12.73. (page 1210)

Use the Binomial Theorem to expand ${(3 x - 2 y)}^{5}$

Short Answer

Expert verified

The expansion of the given binomial is ${(3 x - 2 y)}^{5} = 243 x^{5} - 810 x^{4} y + 1080 x^{3} y^{2} - 720 x^{2} y^{3} + 240 x y^{4} - 32 y^{5}$

Step by step solution

Step 1. Given information.

The given expression is ${(3 x - 2 y)}^{5}$ .

Step 2. Substitute the values and simplify.

Evaluate the coefficients.

Remember,

$a = 3 x b = - 2 y c = 5$

Now put the values,

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

Simplify the exponents

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

Remember that,

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

Simplify the coefficients.

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

Simplify, $243 x^{5} + 5.81 x^{4} (- 2 y) + 10.27 x^{3} (4 y^{2}) + 10.9 x^{2} (- 8 y^{3}) + 5.3 x (16 y^{4}) + (- 32 y^{5})$

Step 3. Conclusion.

Therefore, the expansion of the given expression is ${(3 x - 2 y)}^{5} = 243 x^{5} - 810 x^{4} y + 1080 x^{3} y^{2} - 720 x^{2} y^{3} + 240 x y^{4} - 32 y^{5}$

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Use the Binomial Theorem to expand ${(3 x - 2 y)}^{5}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitute the values and simplify.

Step 3. Conclusion.

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Use the Binomial Theorem to expand3x-2y5

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Substitute the values and simplify.

Step 3. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Geometry

Applied Mathematics

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Statistics

Study anywhere. Anytime. Across all devices.

Use the Binomial Theorem to expand ${(3 x - 2 y)}^{5}$