Problem 51 - Algebra A Combined Function

Question

Multiply. \((7 x y-y)^{2}\)

Step-by-Step Solution

Answer

The expanded form is \(49x^2y^2 - 14xy^2 + y^2\).

Step 1: Understand the Expression

We need to find the square of the binomial expression \((7xy - y)^2\). This means we will be multiplying the binomial by itself.

Step 2: Apply the Binomial Expansion Formula

Use the formula \((a - b)^2 = a^2 - 2ab + b^2\) to expand the expression. Here, \(a = 7xy\) and \(b = y\).

Step 3: Calculate Each Term

1. Compute \(a^2 = (7xy)^2 = 49x^2y^2\).2. Compute \(-2ab = -2(7xy)(y) = -14xy^2\).3. Compute \(b^2 = y^2\).

Step 4: Combine All Terms

Add all the terms from Step 3: \(49x^2y^2 - 14xy^2 + y^2\).

Step 5: Write the Final Expression

The expanded form of \((7xy - y)^2\) is \(49x^2y^2 - 14xy^2 + y^2\).

Key Concepts

Binomial Expansion

The term "binomial expansion" refers to the process of expanding an expression that is squared or raised to a power. A binomial is simply a two-term algebraic expression. For example, in any expression like \[ (a - b)^2 \] , we treat it as multiplying the binomial by itself.
The key formula that helps simplify this is called the binomial expansion formula. For squares, it looks like:

• \( (a - b)^2 = a^2 - 2ab + b^2 \)

This formula helps us avoid lengthy calculations by giving a shortcut to find the final expanded form.
In practical terms, if you have something like \((7xy - y)^2\), then by assigning \(a = 7xy\) and \(b = y\), you quickly find each part of the formula. Using the binomial expansion formula simplifies the calculation process by breaking it down into smaller, manageable parts.

Algebraic Multiplication

Algebraic multiplication involves multiplying terms in algebra, particularly those involving variables as well as numbers. When you multiply binomials, you follow a systematic approach. This helps ensure that each part of the multiplication is considered. In the binomial case, we are doing the following:

• Multiply each term in the first binomial by each term in the second binomial.
• Combine like terms to simplify the expression.

In our example with \((7xy - y)^2\), this multiplication involves:

• Calculating \((7xy)^2\)
• Multiplying \(-2 \cdot 7xy \cdot y\)
• Finding \(y^2\)

Each multiplication step involves dealing with the coefficients and exponents of the variables correctly. This ensures that the expanded expression is both simplified and accurate.

Polynomial Expansion

Polynomial expansion takes the idea of expanding binomials and applies it to larger algebraic expressions called polynomials. A polynomial can have any number of terms, unlike a binomial which has just two.
The expansion of polynomials involves similar steps as binomial expansion but can get increasingly complex with more terms. The goal is to apply distributive laws and simplify each part until you reach a final expanded form.
For example, in a situation like this:

• Simplify each term individually first.
• Combine like terms at the end.

This turns the initial multiplication into a simpler, expanded expression.
In our original exercise, we effectively took the polynomials resulting from \((7xy - y)^2\) and expanded them, combining terms efficiently to reach the final expanded polynomial: \[ 49x^2y^2 - 14xy^2 + y^2 \] Each step in a polynomial expansion requires attention to detail to catch all terms and combine them correctly.