Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ \left(x^{2}-3 y^{2}\right)^{2} $$

Step-by-Step Solution

Answer

The polynomial is \( x^{4} - 6x^{2}y^{2} + 9y^{4} \).

Step 1: Understand the Problem

We need to expand the expression \( \left(x^{2}-3 y^{2}\right)^{2} \). This is a binomial raised to the power of 2, meaning we have to multiply \( x^{2} - 3y^{2} \) by itself.

Step 2: Apply the Binomial Formula

When we expand \( (a-b)^{2} \), it results in \( a^{2} - 2ab + b^{2} \). Here, \( a = x^{2} \) and \( b = 3y^{2} \). We will apply this formula to expand the expression.

Step 3: Calculate Each Term

Calculate each part of the expanded formula:- \( a^{2} = (x^{2})^{2} = x^{4} \).- \( 2ab = 2(x^{2})(3y^{2}) = 6x^{2}y^{2} \).- \( b^{2} = (3y^{2})^{2} = 9y^{4} \).

Step 4: Compile the Expanded Polynomial

Put it all together to form the polynomial: \( x^{4} - 6x^{2}y^{2} + 9y^{4} \).

Key Concepts

Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra that helps in expanding expressions raised to a power. Particularly, it provides a formula for expanding powers of a binomial. A binomial is simply an algebraic expression that consists of two terms, connected by either a plus or minus sign, like \( (x + y) \) or \( (x - y) \). The theorem gives us a systematic way to compute the powers of such binomials. For example, if you have \((a + b)^n\), the theorem allows you to express this as a sum of terms involving powers of \(a\) and \(b\).
For a simple power of 2, as seen in the exercise, the binomial theorem results in \( (a-b)^2 = a^2 - 2ab + b^2 \). This formula is straightforward but lays the base for more complex expansions, where higher powers of a binomial involve additional terms. Understanding the binomial theorem is crucial as it streamlines calculations and unveils patterns in algebraic expressions.

Polynomial Expansion

Polynomial expansion is the process of transforming a compact polynomial expression into a fully expanded form. This expansion involves distributing terms and combining like terms to create a simpler form that is more straightforward to work with. For instance, knowing how to expand a polynomial is vital when dealing with expressions like \((x^2 - 3y^2)^2\), as you turn a compact expression into something more manageable, like \(x^4 - 6x^2y^2 + 9y^4\).
Expanding involves applying the rules of algebra, such as the distributive property, to systematically work through the multiplication of terms. With each step, terms are simplified to reach the final expression. This ensures that each term is accounted for properly, and helps eliminate any potential for error in calculations involving polynomial expressions. It's a skill that, once mastered, can be incredibly powerful and is essential for solving various algebraic problems.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are the essence of algebra and serve as the building blocks for equations and polynomials.
An example would be \((x^2 - 3y^2)\), which is an expression involving two variables \(x\) and \(y\). These expressions can be manipulated and transformed using algebraic rules. Understanding algebraic expressions helps in recognizing and setting up problems so that they can be solved using appropriate mathematical techniques.

• They can be simple, like \(x+2\), or more complex, as seen with polynomials.
• Understanding how to manipulate algebraic expressions is key in simplifying and solving equations.

This manipulation includes expanding, factoring, and simplifying expressions using a set of defined algebraic rules. As with the binomial theorem and polynomial expansion, creating clarity from complexity is what makes algebraic expressions intriguing and invaluable.