Problem 32 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using a Special Product Formula and simplify. $$(x-3 y)^{2}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Special Product Formula

The expression given is \((x - 3y)^2\). This is the "square of a binomial" form, which is expressed by the formula \((a - b)^2 = a^2 - 2ab + b^2\). In this case, let \(a = x\) and \(b = 3y\).

Step 2: Apply the Special Product Formula

Using the formula \((a - b)^2 = a^2 - 2ab + b^2\), substitute \(a = x\) and \(b = 3y\) into the formula. This gives\[ (x - 3y)^2 = x^2 - 2(x)(3y) + (3y)^2. \]

Step 3: Calculate Each Term

Now, calculate each term in the expression:- First Term: \(x^2\) remains as \(x^2\).- Second Term: \(-2(x)(3y) = -6xy\).- Third Term: \((3y)^2 = 9y^2\).

Step 4: Combine the Terms

Combine the calculated terms into the final expression:\[ x^2 - 6xy + 9y^2. \]

Key Concepts

Binomial Expansion

When expanding a binomial, like \(x - 3y\)^2, using a special product formula can simplify the process significantly. Binomial expansion involves breaking down expressions of the form \(a + b\)^n or \(a - b\)^n, where 'a' and 'b' are any algebraic expressions, and 'n' is a non-negative integer. In the square of a binomial case, \(a - b\)^2 can be expanded using the formula \(a^2 - 2ab + b^2\). This formula allows for a quick and efficient way to expand the expression without manually multiplying each term.

In our exercise, we specifically used the formula for \(a - b\)^2 to expand \(x - 3y\)^2. Identifying 'a' as \x\ and 'b' as \3y\ means plugging these into the formula to simplify the process. This step ensures accuracy and saves time compared to expanding each term independently. Utilizing the binomial expansion formula not only ensures correctness but also improves understanding of polynomial expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition or multiplication). These combinations are essential in mathematics as they represent quantities and relationships. An expression like \(x - 3y\)^2 consists of two terms: \x\ and \3y\. When dealing with algebraic expressions, it's important to understand how these terms interact, especially when operations like squaring or expanding are involved.

The structure of an algebraic expression involves terms, coefficients, variables, and sometimes exponents. Coefficients are the numerical parts that multiply the variables; here, -3 is the coefficient of \y\. Operations such as addition, subtraction, multiplication, and division help in manipulating these expressions to simplify or solve them. In the example provided, squaring the expression means multiplying it by itself, and using known formulas helps in making this task more manageable.

Polynomial Simplification

Simplifying polynomials involves reducing expressions to their simplest form, combining like terms, and ensuring no further simplification is possible. In our case with \(x - 3y\)^2, applying the binomial expansion formula results in each term being calculated separately: \x^2\, -6xy, and \9y^2\.

The simplification process led to a polynomial expression, \x^2 - 6xy + 9y^2\, where each term has been expanded and combined. Simplification makes calculations in algebra easier and expressions clearer, especially when assessing or solving for variables. Recognizing and applying strategies like these is key to mastering algebraic manipulation. Simplification ensures accuracy and better analysis of mathematical relationships within polynomials.