Problem 2 - Algebra for College Students

Question

Find each indicated product. Remember the shortcut for multiplying binomials and the other special patterns we discussed in this section. $$3 x^{2} y\left(6 y^{2}-5 x^{2} y^{4}\right)$$

Step-by-Step Solution

Answer

The product is \(18x^2y^3 - 15x^4y^5\).

Step 1: Distribute the First Term

Multiply the first term outside the parenthesis, which is \(3x^2y\), by each term inside the parenthesis separately. First, multiply \(3x^2y\) by \(6y^2\). This gives: \(3x^2y imes 6y^2 = 18x^2y^3\).

Step 2: Distribute the First Term Again

Now multiply \(3x^2y\) by the second term inside the parenthesis, which is \(-5x^2y^4\). This gives: \(3x^2y imes -5x^2y^4 = -15x^{4}y^{5}\).

Step 3: Combine the Results

After distributing, combine the results from both multiplications to get the final expanded form of the expression: \(18x^2y^3 - 15x^4y^5\).

Key Concepts

Distributive Property

The distributive property is a key concept in algebra that helps us simplify expressions. It tells us how to multiply a single term by a sum or difference inside parentheses.
This property is defined as:

• For any numbers or expressions, it states that \( a(b + c) = ab + ac \).

This means you take the term outside the parenthesis and multiply it by each term inside the parenthesis separately.
In the context of our exercise, the term outside the parenthesis is \( 3x^2y \). We applied the distributive property by multiplying this term with each term inside the parenthesis: \( 6y^2 \, \) and \( -5x^2y^4 \). The results were combined to expand the expression fully.
Using the distributive property correctly can significantly simplify polynomial multiplication and ensure you arrive at correct results quickly.

Binomials

Binomials are a special type of polynomial expression. They consist of exactly two terms connected either by addition or subtraction.
An example might be \( a + b \) or \( x - y \). In the realm of our exercise, the expression \( 6y^2 - 5x^2y^4 \) is a binomial as it has two distinct terms.

• The first term is \( 6y^2 \)
• The second term is \( -5x^2y^4 \)

When multiplying a monomial (single term) by a binomial, we use the distributive property, distributing the monomial across each term in the binomial to produce new, simplified terms.
In our case, each multiplication involving \( 3x^2y \) gives a result that contributes to the final expanded polynomial.

Algebraic Expressions

An algebraic expression is a mathematical phrase that contains numbers, variables, and operators.
These expressions are fundamental in algebra for representing relationships and calculations. Our exercise featured an expression like \( 3x^2y(6y^2 - 5x^2y^4) \), which contains:

• Coefficients: numerical factors, like 3 and 6.
• Variables: symbols that can represent numbers, such as \( x \) and \( y \).
• Exponents: powers that indicate repeated multiplication, such as \( x^2 \) and \( y^4 \).

To solve or simplify these expressions, understanding how to apply operations like multiplication and distribution is essential.
By recognizing the structure of algebraic expressions, we can use techniques such as polynomial multiplication effectively. This not only helps in finding products like our exercise's example but also in solving equations and understanding algebra more broadly.