Question

Verify each expansion. Obtain the binomial coefficients by formula or from Pascal's triangle as directed by your instructor. $$\left(2 a^{2}+\sqrt{b}\right)^{5}=32 a^{10}+80 a^{8} b^{1 / 2}+80 a^{6} b+40 a^{4} b^{3 / 2}+10 a^{2} b^{2}+b^{5 / 2}$$

Short answer

The expansion \( (2a^2 + \sqrt{b})^5 \) is correct with the given coefficients.

Step-by-step solution

Understand the Binomial Theorem

The expansion of \( (x + y)^n \) can be written using the Binomial Theorem, which is \( \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k \) where \( {n \choose k} \) is a binomial coefficient that can be obtained from Pascal's triangle or calculated using the formula \( {n \choose k} = \frac{n!}{k!(n-k)!} \) where \( n! \) is the factorial of \( n \) and \( ! \) denotes the factorial operation.

Calculate Binomial Coefficients

For the expansion of \( (2a^2 + \sqrt{b})^5 \) we need to find the binomial coefficients for \( n = 5 \) which are \( {5 \choose 0}, {5 \choose 1}, {5 \choose 2}, {5 \choose 3}, {5 \choose 4}, {5 \choose 5} \) using the formula or Pascal's triangle. We obtain the coefficients 1, 5, 10, 10, 5, 1 respectively.

Apply the Binomial Coefficients

Using the calculated coefficients, we can expand \( (2a^2 + \sqrt{b})^5 \) as \((1)(2a^2)^5 + (5)(2a^2)^4(\sqrt{b}) + (10)(2a^2)^3(\sqrt{b})^2 + (10)(2a^2)^2(\sqrt{b})^3 + (5)(2a^2)(\sqrt{b})^4 + (1)(\sqrt{b})^5\) to obtain the terms of the expansion.

Simplify the Expansion Terms

Simplify each term of the expansion: \[32a^{10}\] for the first term, \[80a^8b^{\frac{1}{2}}\] for the second term, \[80a^6b\] for the third term, \[40a^4b^{\frac{3}{2}}\] for the fourth term, \[10a^2b^2\] for the fifth term, and \[b^{\frac{5}{2}}\] for the sixth term.

Combine All Terms for the Final Answer

Combine all the simplified terms to write down the expanded form \(32 a^{10}+80 a^{8} b^{1 / 2}+80 a^{6} b+40 a^{4} b^{3 / 2}+10 a^{2} b^{2}+b^{5 / 2}\). This confirms that the given expansion is correct.

Key concepts

Binomial Coefficients

When expanding expressions using the binomial theorem, an essential component is the binomial coefficients. These coefficients are the numbers that appear in the expanded form of a binomial expression like \( (x+y)^n \). Binomial coefficients are noted as \( {n \choose k} \), where \({n \choose k} \) represents the coefficient of the k-th term in the expansion of \( (x+y)^n \).

For calculating the binomial coefficients, we use the formula \( {n \choose k} = \frac{n!}{k!(n-k)!} \), where \(!\) represents the factorial operation, meaning the product of all positive integers up to that number. For example, \(3! = 3 \times 2 \times 1 = 6\).

For simplicity and to avoid lengthy calculations, these coefficients can also be directly obtained from Pascal's triangle. In the context of the provided exercise, the binomial coefficients for \( (2a^2 + \sqrt{b})^5 \) are \(1, 5, 10, 10, 5, 1\), corresponding to the terms from \( {5 \choose 0} \) to \( {5 \choose 5} \) inclusively.

Pascal's Triangle

A convenient way to visualize binomial coefficients is through Pascal's triangle. This triangle is constructed by starting with a '1' at the top, and every number below it is the sum of the two numbers directly above it. The n-th row of Pascal's triangle gives the binomial coefficients for the expansion of \( (x+y)^n \).

To use Pascal's triangle for our problem, look at the row starting with 1, 5, which represents the coefficients for \( n=5 \). The numbers in this row are exactly the binomial coefficients used in the expansion of \( (2a^2 + \sqrt{b})^5 \), thus providing a quick reference without the need for calculation using the factorial-based formula for each coefficient.

Factorial Operation

The factorial operation is pivotal in understanding the binomial coefficients and thus affects the binomial theorem and polynomial expansions. A factorial, represented as \(n!\), is the product of all positive integers from 1 to n. For example, \(4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24\).

Factorials increase very quickly with n, and therefore computational tools may be needed for large values of n. In the binomial theorem, the coefficients are found using factorials, as seen in the formula \( {n \choose k} = \frac{n!}{k!(n-k)!} \). It's important to remember that the factorial of zero is defined as \(0! = 1\) for the purposes of these calculations.

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial raised to a power as a sum of terms. Using the binomial theorem, we can expand a binomial expression \( (x+y)^n \) into a polynomial with \( n+1 \) terms. The general form of this expansion is given by the summation \( \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k \), where each term corresponds to a specific binomial coefficient and the powers of x and y.

In practice, like our exercise with \( (2a^2 + \sqrt{b})^5 \), the polynomial expansion involves determining the binomial coefficients (either through Pascal's triangle or the formula based on factorials) and applying those coefficients to each term in the series. The final polynomial is then the sum of all these terms, which are simplified accordingly to derive at the expanded expression.