Chapter 11: Q. 12.74. (page 1069)

Use the Binomial Theorem to expand ${(4 x - 3 y)}^{4}$ .

Short Answer

Expert verified

The expansion of the given binomial is ${(4 x - 3 y)}^{4} = 256 x^{4} - 768 x^{3} y + 864 x^{2} y^{2} - 432 x y^{3} + 81 y^{4}$

Step by step solution

Step 1. Substitute the values.

The given expression is ${(4 x - 3 y)}^{4}$ .

Substitute in the values into the binomial expansion,

$a = 4 x b = - 3 y n = 4$

Substitute the values.

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

Simplify the exponents.

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

Step 2. Simplify the coefficients.

Evaluate the coefficients. Remember,

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

Simplify the exponents.

localid="1668001816525" ${(4 x - 3 y)}^{4} = 256 x^{4} + 4 \cdot 64 x^{3} \cdot (- 3 y) + \frac{4!}{2! 2!} (16 x^{2}) \cdot (9 y^{2}) + \frac{4!}{3! 1!} (4 x) (- 27 y^{3}) + 81 y^{4}$

Simplify,

localid="1668001843414" $256 x^{4} - 768 x^{3} y + 6 \cdot (16 x^{2}) \cdot (9 y^{2}) + 4 \cdot (4 x) (- 27 y^{3}) + 81 y^{4}$

Step 3. Conclusion.

Therefore, the expansion of the given expression is,

${(4 x - 3 y)}^{4} = 256 x^{4} - 768 x^{3} y + 864 x^{2} y^{2} - 432 x y^{3} + 81 y^{4}$

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Use the Binomial Theorem to expand ${(4 x - 3 y)}^{4}$ .

Short Answer

Step by step solution

Step 1. Substitute the values.

Step 2. Simplify the coefficients.

Step 3. Conclusion.

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Use the Binomial Theorem to expand4x-3y4.

Short Answer

Step by step solution

Step 1. Substitute the values.

Step 2. Simplify the coefficients.

Step 3. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

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Study anywhere. Anytime. Across all devices.

Use the Binomial Theorem to expand ${(4 x - 3 y)}^{4}$ .