Problem 56 - Intermediate Algebra

Question

Multiply. See Example 4 $$25\left(2 a^{2}-3 a+1\right)$$

Step-by-Step Solution

Answer

The product is \(50a^2 - 75a + 25\).

Step 1: Apply the Distributive Property

We begin by using the distributive property to distribute the 25 across the terms inside the parenthesis. The expression is: \[25(2a^2 - 3a + 1).\]This results in multiplying 25 with each term inside the parenthesis.

Step 2: Multiply Each Term

Compute the multiplication of 25 with each term individually:- For the first term: \(25 \times 2a^2 = 50a^2\).- For the second term: \(25 \times (-3a) = -75a\).- For the third term: \(25 \times 1 = 25\).

Step 3: Combine the Results

Combine all the terms from the previous step to create the final expanded expression:\[50a^2 - 75a + 25.\]

Key Concepts

Understanding Polynomial Multiplication

Polynomial multiplication is an important operation in algebra that involves multiplying two or more polynomials together. One fundamental mechanism used is the distributive property. This property states that for any numbers or variables, \(a(b + c) = ab + ac\). When applied to polynomials, this means each term in one polynomial must be multiplied by each term in the other.

In the original exercise, you applied this principle by multiplying each term inside the parenthesis with 25. This method ensures that every part of one polynomial is appropriately enlarged by every part of the other. The outcome is a new polynomial with its terms expanded.

When multiplying polynomials, it's crucial to pay attention to each of the coefficients and exponents to ensure accurate results.

• Identify each term in both polynomials.
• Multiply them using the distributive property.
• Combine like terms to simplify.

Polishing these skills is key to mastering algebraic expressions and efficiently managing polynomial multiplication.

Exploring Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operational symbols but do not have an equality sign. They are the building blocks of algebra. In our exercise, the expression is inside parenthesis and includes terms like \(2a^2\), \(-3a\), and \(1\).

These expressions allow us to represent real-world problems in an abstract mathematical format, which can then be manipulated and solved using various rules and principles like the distributive property.

An essential part of working with algebraic expressions is understanding:

• **Terms:** Individual parts of an expression separated by plus or minus signs.
• **Coefficients:** Numbers multiplied by variables (e.g., in \(2a^2\), 2 is the coefficient).
• **Variables:** Letters that represent unknown values.

By dissecting the parts of an algebraic expression, one can better understand and manipulate them to produce valid solutions to algebra problems.

The Role of Mathematical Operations in Algebra

Mathematical operations, such as addition, subtraction, multiplication, and division, form the backbone of working with algebraic expressions. Each operation follows specific rules and properties, crucial for conducting accurate calculations. In our example, multiplication was the primary focus.

When performing these operations on polynomials, especially multiplication, it is vital to maintain the integrity of the operation rules. The distributive property is prominent here as it dictates how multiplication distributes over addition or subtraction inside expressions.

• **Addition and Subtraction:** Used to combine like terms after multiplication.
• **Multiplication:** Amplifies terms, often resulting in higher-degree polynomials.

Effective management of these operations produces correct polynomial expressions from the original form. Grasping these fundamental operations is essential for anyone looking to excel in algebra and other higher-level mathematics.