Problem 50 - Algebra 2

Question

Simplify. $$ (x-3 y)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(x^2 - 6xy + 9y^2\).

Step 1: Write the Expression

We start with the expression \((x-3y)^2\). Our goal is to expand this expression to simplify it.

Step 2: Recognize the Binomial Square

The expression \((x-3y)^2\) is in the form of a squared binomial \((a - b)^2\), which can be expanded using the formula: \((a - b)^2 = a^2 - 2ab + b^2\). Here, \(a = x\) and \(b = 3y\).

Step 3: Expand the Expression

Apply the expansion formula: \((x-3y)^2 = x^2 - 2(x)(3y) + (3y)^2\).

Step 4: Simplify Each Term

Calculate each term individually: - The first term is \(x^2\).- The second term is \(-2(x)(3y) = -6xy\).- The third term is \((3y)^2 = 9y^2\).

Step 5: Combine the Terms

Combine all of the expanded terms to form the simplified expression: \(x^2 - 6xy + 9y^2\).

Key Concepts

Squared Binomial

A squared binomial is an algebraic expression formed by squaring a binomial. A binomial is simply a polynomial comprised of two terms, connected by either addition or subtraction. When you square a binomial, you are multiplying the binomial by itself. For example, the expression

• \((x - 3y)^2\) represents the square of the binomial \((x - 3y)\).

Using the formula for a squared binomial:

• \((a - b)^2 = a^2 - 2ab + b^2\),

you can expand the expression to form a quadratic trinomial. This transformation helps in simplifying the expression and is a fundamental technique in algebra.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and operation symbols like addition, subtraction, multiplication, and division. These expressions do not have an equality sign as equations do.In the expression \((x - 3y)^2\):

• \(x\) and \(y\) are variables representing unknown values.
• The term \(3y\) shows multiplication between a number and a variable.

The operations involved include both subtraction and exponentiation. Understanding and correctly interpreting algebraic expressions is vital for expanding, factoring, and simplifying expressions in algebra. Each part of the expression plays a role in defining its value and form.

Polynomial Simplification

Polynomial simplification involves carrying out all possible arithmetic operations to reduce the expression to its simplest form. This often includes expanding polynomials and combining like terms to make the expression as straightforward as possible.For the squared binomial \((x - 3y)^2\), the simplification process involves:

• Expanding the binomial using the formula \((a-b)^2 = a^2 - 2ab + b^2\), which results in \(x^2 - 6xy + 9y^2\).
• Calculating each term: \(x^2\), \(-6xy\), and \(9y^2\) using basic arithmetic operations.
• Combining these terms into the final simplified form.

This approach allows one to work with simpler expressions, making further calculations or analysis easier and clearer.