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Chapter 14: Problem 18

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$\left(x^{2}+y\right)^{4}$$

Short Answer

Expert verified
\(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\)

Step by step solution

01

Identify the elements and the power in the binomial

Here, the binomial given is \((x^{2}+y)\) and the power to which it is raised is \(4\). Thus in our binomial theorem, \(a = x^2, b = y, n = 4\).
02

Apply the binomial theorem

Next, we will apply the binomial theorem to expand the binomial expression. The expression becomes: \[\binom{4}{0}(x^2)^4y^0 + \binom{4}{1}(x^2)^3y^1 + \binom{4}{2}(x^2)^2*y^2 + \binom{4}{3}(x^2)^1*y^3 + \binom{4}{4}(x^2)^0*y^4\]. Here \(\binom{4}{k} = \frac{4!}{k!(4-k)!}\) where \( ! \) denotes factorial.
03

Calculate coefficients and simplify the expression

We will calculate the binomial coefficients and simplify each term to get the final expression. The final expression becomes: \(x^8 + 4x^6y + 6x^4y^2 + 4x^2y^3 + y^4\) after calculating coefficients and simplifying.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I used a formula to find the sum of the infinite geometric series \(3+1+\frac{1}{3}+\frac{1}{9}+\cdots\) and then checked my answer by actually adding all the terms.

Use the formula for the value of an annuity to solve Exercises. Round answers to the nearest dollar. To offer scholarship funds to children of employees, a company invests \(\$ 10,000\) at the end of every three months in an annuity that pays \(10.5 \%\) compounded quarterly. a. How much will the company have in scholarship funds at the end of ten years? b. Find the interest.

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Explain how to find the general term of a geometric sequence.

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