Problem 35 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ \left(2 x^{2}+3 y^{2}\right)^{2} $$

Step-by-Step Solution

Answer

\(4x^4 + 12x^2y^2 + 9y^4\)

Step 1: Apply the Binomial Theorem

According to the binomial theorem, we can expand the square of a binomial as follows: \((a + b)^2 = a^2 + 2ab + b^2\). In this case, \(a = 2x^2\) and \(b = 3y^2\).

Step 2: Calculate \(a^2\)

Square the first term, \(a = 2x^2\): \( (2x^2)^2 = 4x^4 \).

Step 3: Calculate \(2ab\)

Multiply the two terms and then multiply by 2: \(2 \cdot (2x^2) \cdot (3y^2) = 12x^2y^2\).

Step 4: Calculate \(b^2\)

Square the second term, \(b = 3y^2\): \((3y^2)^2 = 9y^4\).

Step 5: Combine the Terms

Add all the results from steps 2, 3, and 4 to get the expanded form: \(4x^4 + 12x^2y^2 + 9y^4\). Simplify if necessary.

Key Concepts

Binomial Theorem

The Binomial Theorem is a cornerstone of algebra, offering a way to expand expressions raised to a power. In this context, it allows us to expand \( (a + b)^2 \) quickly without manually writing out and multiplying terms by themselves. The theorem states that \( (a + b)^2 = a^2 + 2ab + b^2 \).

• \(a^2\) is simply the square of the first term.
• \(b^2\) is the square of the second term.
• \(2ab\) represents two times the product of the first and second term.

By applying this theorem, complex polynomial expressions become manageable. The key takeaway is that it provides a precise pattern for expansion, which can be generalized to higher powers.

Algebraic Expressions

Algebraic expressions are essentially mathematical phrases that include numbers, variables, and operational signs. Understanding the structure and components of these expressions is crucial for applying algebraic methods effectively.In the given problem, the expression is \( (2x^2 + 3y^2)^2 \). This expression involves:

• Terms: Components separated by addition or subtraction. Here, \(2x^2\) and \(3y^2\) are terms.
• Coefficients: Numbers in front of the variables. In \(2x^2\), 2 is the coefficient.
• Variables and Exponents: Letters representing numbers, and exponents indicating repeated multiplication. \(x^2\) means \(x\) is multiplied by itself.

Recognizing these allows for manipulation and proper application of the binomial theorem to simplify expressions.

Polynomial Simplification

Polynomial simplification involves making an expression easier to work with while still expressing the same mathematical relationship. This process is vital in algebra to reduce complexity and reveal the essential structure of expressions.In the exercise, we simplify the expression \( (2x^2 + 3y^2)^2 \) by using the binomial theorem, resulting in \( 4x^4 + 12x^2y^2 + 9y^4 \). Simplification includes steps such as:

• Combining like terms: Though not needed here, in some cases terms are combined if they share the same variable factor.
• Reducing expressions: Breaking terms down or rearranging them for clarity.

Effective simplification not only makes the expression tidier but also prepares it for further algebraic operations.