Problem 70 - College Algebra

Question

Find each product. $$(3 x-y)(2 x+5 y)$$

Step-by-Step Solution

Answer

The product of the polynomials \(3x - y\) and \(2x + 5y\) is \(6x^2 + 13xy - 5y^2\).

Step 1: Distribute the first term of the first polynomial

Multiply \(3x\) of the first polynomial with every term of the second polynomial: \((3x)*(2x) = 6x^2\), \((3x)*(5y) = 15xy\).

Step 2: Distribute the second term of the first polynomial

Multiply \(-y\) of the first polynomial with every term of the second polynomial: \((-y)*(2x) = -2xy\), \((-y)*(5y) = -5y^2\).

Step 3: Combine all terms

Add the results of the two steps: \(6x^2 + 15xy - 2xy - 5y^2 = 6x^2 + 13xy - 5y^2\)

Key Concepts

Distributive Property

At the heart of polynomial multiplication, we have the distributive property. This algebraic principle is what enables us to multiply a single term by each term within a polynomial. Think of the distributive property as an efficient way to eliminate parentheses in expressions.

For example, when we look at the exercise \( (3x - y)(2x + 5y) \), we apply the distributive property twice. Each term in the first polynomial is distributed, or multiplied, by each term in the second polynomial. This gives us \(3x * 2x = 6x^2\) and \(3x * 5y = 15xy\) for the first term; \( - y * 2x = -2xy\) and \( - y * 5y = -5y^2\) for the second term. It is essential to maintain the order of operations and be mindful of the signs. If you struggle with keeping track of positive and negative signs during distribution, a tip would be to treat them like numbers and perform operations as usual.

Combining Like Terms

Once the distributive property has been applied and all the terms are multiplied out, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same power, even if they have different coefficients.

In our example, after applying the distributive property, we end up with \(6x^2\), \(15xy\), \( -2xy\), and \( -5y^2\). Here, \(15xy\) and \( -2xy\) are like terms, so we combine them by adding their coefficients: \(15 - 2 = 13\), resulting in \(13xy\). The expression simplifies to \(6x^2 + 13xy - 5y^2\). The art of combining like terms is a critical skill in algebra that helps reduce complex expressions into simpler forms.

Algebraic Expressions

Understanding algebraic expressions is foundational to mastering polynomial multiplication. An algebraic expression is a combination of numbers, variables (like \(x\) and \(y\)), and arithmetic operations like addition, subtraction, multiplication, and division.

In the context of our exercise, \(3x\) and \( - y\) are terms of the first algebraic expression, while \(2x\) and \(5y\) comprise the second. When we speak of terms, it refers to individual parts of an expression separated by plus or minus signs.

While working with algebraic expressions, remember to respect the conventional order of operations and manipulation rules, like the distributive property and combining like terms, as we've explored. This holistic understanding will lead to success in algebra and beyond, turning complex problems into manageable tasks.