Problem 39 - College Algebra and Calculus: An Applied Approach

Question

Find the product.\([(x-3)+y]^{2}\)

Step-by-Step Solution

Answer

The product of \([(x-3)+y]^{2}\) when expanded is \(x^{2}+2xy+y^{2}-6x-6y+9\)

Step 1: Identify 'A' and 'B'

In this binomial, \(A=(x-3)\) and \(B=y\). We will isolate these variables because they will be used to expand the expression.

Step 2: Substitute 'A' and 'B' into the binomial formula

The standard formula for a binomial squared is \(A^{2}+2AB+B^{2}\). Substitute \(A=(x-3)\) and \(B=y\) into the formula. This gives us \(((x-3)^{2}+2(x-3)y+y^{2})\).

Step 3: Simplify the expression

Now, expand and simplify the expression. First, expand \((x-3)^{2}\) to \(x^{2}-6x+9\). Then, expand \(2(x-3)y\) to \(2xy-6y\). Lastly, \(y^{2}\) remains the same. The simplified result is then \(x^{2}+2xy-6x+y^{2}-6y+9\).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful algebraic technique used to expand expressions that are raised to a power, like \( (x - 3 + y)^2 \). When you see a squared term in this form, it hints at the presence of a simple pattern governed by the Binomial Theorem.
This theorem states that \( (A + B)^n \) can be expanded into a sum involving terms of the form \( \binom{n}{k} A^{n-k} B^k \). For squaring, which means \( n = 2 \), this becomes \( A^2 + 2AB + B^2 \).

• First recognize \( A \) and \( B \) in your expression, here \( A = (x-3) \) and \( B = y \).
• Using the formula, \( (x-3)^2 + 2(x-3)(y) + y^2 \), you get the expanded form.

This pattern is straightforward but incredibly useful for simplifying polynomial expressions.

Polynomial Expressions

Polynomial expressions are algebraic expressions made up of variables and coefficients, involving operations of addition, subtraction, and multiplication. In the context of binomial expansions, understanding polynomials is crucial.
For example, when expanding \( (x-3)^2 \), you perform a multiplication, resulting in a polynomial \( x^2 - 6x + 9 \).
Polynomials can consist of:

• Monomials, like \( x^2 \) or \( 9 \)
• Binomials, which have two terms, such as \( 2xy - 6y \)
• And more complex forms, like the entire expression \( x^2 + 2xy - 6x + y^2 - 6y + 9 \)

Understanding their structure helps in operations like addition, subtraction, and particularly, expansions.

Algebraic Simplification

Algebraic simplification involves rewriting expressions in a simpler or more usable form. The goal is to make a complex expression easier to understand and work with.
In binomial expansions such as \( (x-3+y)^2 \), once expanded, each part of the expression needs to be simplified:

• First, expand \( (x-3)^2 \) into \( x^2 - 6x + 9 \).
• Next, consider \( 2(x-3)y \), which becomes \( 2xy - 6y \).
• Finally, ensure \( y^2 \) is included.

Combine all terms: \( x^2 + 2xy - 6x + y^2 - 6y + 9 \).
By identifying like terms or common factors, you can simplify further, resulting in a clearer and more manageable expression.