Problem 72 - College Algebra with Modeling and Visualization

Question

Multiply the polynomials. $$(x-y+5)(x y)$$

Step-by-Step Solution

Answer

The product is \(x^2y - y^2x + 5xy\).

Step 1: Distribute the first term

Begin by distributing the first term of the polynomial \((x-y+5)\), which is \(x\). Multiply \(x\) by each term in the second polynomial \((xy)\) to get the expression: \(x \cdot xy = x^2y\).

Step 2: Distribute the second term

Now distribute the second term \((-y)\) of the first polynomial across the second polynomial \((xy)\). Multiply \(-y\) by \(xy\) to obtain: \(-y \cdot xy = -y^2x\).

Step 3: Distribute the third term

Next, distribute the last term \(+5\) of the first polynomial by multiplying across \(xy\). This results in: \(5 \cdot xy = 5xy\).

Step 4: Combine the results

Add all the terms obtained from the distribution steps: \(x^2y\), \(-y^2x\), and \(5xy\). The combined polynomial expression is: \(x^2y - y^2x + 5xy\).

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that allows you to multiply a single term by terms within parentheses. In polynomial multiplication, it helps break down complex expressions into more manageable parts. Imagine you have two groups of numbers, and you need to apply something (like multiplication) to each number in those groups separately and then combine the results.
For example, in the expression

• (x-y+5)(xy),

we treat each term in the first polynomial as a separate entity that must be multiplied by each term in the second polynomial. This systematic breakdown of the expression simplifies the calculation and ensures that no terms are overlooked.
The property can be summarized as:

• If you have an expression of the form \(a(b+c)\), it becomes \(ab + ac\).

In polynomials, every term from the first polynomial multiplies every term in the second. Using the Distributive Property helps ensure accuracy and efficiency in polynomial calculations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They serve as the foundation of algebra and are used to represent mathematical relationships. Variables like \(x\) and \(y\) are typically used to indicate unknown or varying values.
In the exercise, the algebraic expression

• (x-y+5)(xy)

consists of two polynomial units that need to be multiplied. An expression can encompass more than just simple numbers—it includes terms, which are separated by addition or subtraction within an expression. Terms can be constants, variables, or combinations of both.
Understanding the concept of algebraic expressions is crucial because it forms the basis of solving problems in algebra. These expressions make it possible to simplify, evaluate, and manipulate problems to find unknowns. Recognizing how to work with algebraic expressions is an essential skill that aids in breaking down and tackling more complex algebraic problems.

Polynomial Expressions

Polynomial expressions are a type of algebraic expression that consist of terms with non-negative integer exponents. They are essential in various mathematical fields, including calculus and algebra. A polynomial can have one or more terms, and each term is composed of a coefficient and one or more variables raised to an exponent.
For instance, in

• (x-y+5),

we have a polynomial of three terms: \(x\), \(-y\), and \(+5\). Each of these is a component of a larger expression that can be manipulated using the rules of algebra.
The process of multiplying polynomials, as shown in the exercise, involves using each term in the first polynomial to multiply every term in the second, using the Distributive Property. Once this multiplication is complete, the resulting terms are combined to simplify the polynomial expression. This step-by-step multiplication ensures that all resulting terms are accounted for, leading to an accurate outcome.
Polynomials can vary greatly in complexity, but understanding their structure and how to operate with them is key to mastering algebra.