Problem 18 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ \left(2 a^{3}+3 b^{2}\right)^{2} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the algebraic expression \((2a^3 + 3b^2)^2\) is \(4a^6 + 12a^3b^2 + 9b^4\).

Step 1: Apply the binomial theorem to expand the square

We have \((2a^3 + 3b^2)^2 = (2a^3 + 3b^2)(2a^3 + 3b^2)\). To expand this, we can use the foil method, which consists of multiplying the first terms, the outer terms, the inner terms, and the last terms.

Step 2: Multiply the first terms

Multiplying the first terms, we have \((2a^3)(2a^3) = 4a^6\).

Step 3: Multiply the outer terms

Multiplying the outer terms, we have \((2a^3)(3b^2) = 6a^3b^2\).

Step 4: Multiply the inner terms

Multiplying the inner terms, we have \((3b^2)(2a^3) = 6a^3b^2\).

Step 5: Multiply the last terms

Multiplying the last terms, we have \((3b^2)(3b^2) = 9b^4\).

Step 6: Combine the results of steps 2-5

We can now combine the results from steps 2-5 to get the expanded expression: \(4a^6 + 6a^3b^2 + 6a^3b^2 + 9b^4\).

Step 7: Group and simplify like terms

Combining the like terms, we get \(4a^6 + 12a^3b^2 + 9b^4\). The simplified expression is: \((2a^3 + 3b^2)^2 = 4a^6 + 12a^3b^2 + 9b^4\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that helps us expand expressions that are raised to a power, typically involving two terms. When you encounter an expression like \((a + b)^n\), the theorem provides a formula for expanding it without having to multiply the whole expression repeatedly. This is particularly useful for higher powers, where direct multiplication becomes cumbersome.

The horizontal expansion of a binomial expression like \((x+y)^2\) involves neatly laying out its components using the binomial theorem. For our specific problem \((2a^3 + 3b^2)^2\), you can see that we're multiplying the entire expression by itself. The binomial theorem tells you that the coefficients you obtain from this expansion are the binomial coefficients, commonly found in Pascal's Triangle. These coefficients lead us to the terms in the expanded form.

The general formula for the expansion of a binomial \((a + b)^n\) is given by:

• \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \).

For this exercise, where the exponent is 2, the binomial theorem triggers the use of simple multiplication processes — often executed through the use of the FOIL Method.

FOIL Method

The FOIL method is a straightforward technique for multiplying two binomials. It stands for First, Outer, Inner, Last, referring to the terms we multiply together in this specific order.

The FOIL method can easily be learned with practice, by remembering its core steps. Let's use it on the expression \((2a^3 + 3b^2)^2\), which becomes \((2a^3 + 3b^2)(2a^3 + 3b^2)\) \( \text \).

• First: Multiply the first terms in each binomial: \((2a^3)(2a^3) = 4a^6\).
• Outer: Multiply the outer terms: \((2a^3)(3b^2) = 6a^3b^2)\).
• Inner: Multiply the inner terms: \((3b^2)(2a^3) = 6a^3b^2\).
• Last: Multiply the last terms: \((3b^2)(3b^2) = 9b^4\).

By summing these results, you follow through to obtain the expanded polynomial form. Simplification follows suit, as seen in the step-by-step solution.

Polynomial Expansion

Polynomial Expansion refers to the process of transforming a compacted expression into a longer form by carrying out multiplication operations like those demonstrated in using the FOIL method or the Binomial Theorem.

Expanding a polynomial helps to clearly see all the terms that form the entire expression. It is particularly handy when simplifying expressions or solving equations that involve polynomials.

In our exercise, through polynomial expansion, we expanded \((2a^3 + 3b^2)^2\) to finally achieve \(4a^6 + 12a^3b^2 + 9b^4\). Here's the step-by-step procedure:

• First, employ the FOIL method to multiply the binomial terms together.
• Combine like terms such as \(6a^3b^2 + 6a^3b^2\) yielding \(12a^3b^2\).
• Rewrite the expression with the combined terms as a simplified, single polynomial.

Polynomial expansion is fundamental for understanding the structure and factors of algebraic expressions, making it easier to manipulate and solve them.