Problem 4 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \((2 a-4)\left(3 a^{2}+5 a-1\right)\) is the product of a _____ and a _____.

Step-by-Step Solution

Answer

The expression is the product of a binomial and a trinomial.

Step 1: Identify the Expression's Structure

Look at the given expression \((2a - 4)(3a^2 + 5a - 1)\). This expression involves the product of two sections separated by a multiplication operation.

Step 2: Determine the First Blank

Examine the first section, \(2a - 4\). It consists of two terms, where the highest power of the variable \(a\) is 1. Therefore, it is a binomial.

Step 3: Determine the Second Blank

Examine the second section, \(3a^2 + 5a - 1\). It has three terms, with the highest power of \(a\) being 2, indicating it is a trinomial.

Step 4: Formulate the Final Answer

Based on the analysis, \((2a - 4)\) is a binomial, and \(3a^2 + 5a - 1\) is a trinomial, confirming the expression as a product of a binomial and a trinomial.

Key Concepts

Binomial

A binomial is a specific type of polynomial that contains exactly two distinct terms. These terms are usually separated by a plus or minus sign. Each term can have variables or just numbers, but the key characteristic is the presence of two separate parts.

In the example from the exercise, the expression \(2a - 4\) is a binomial. We know this because it has two terms: \(2a\) and \(-4\). The highest power of the variable \(a\) in any of these terms is 1, making it a linear binomial.

A few points to consider when identifying binomials:

• Count the terms: A binomial always has exactly two terms.
• Watch for different operations: Terms can be combined with addition or subtraction.
• Check the exponents: Binomials can have various degrees, but each term will contribute to understanding the polynomial's degree.

Recognizing binomials is crucial in algebra as they frequently appear in expressions and equations.

Trinomial

Trinomials are another kind of polynomial but, unlike binomials, they consist of three terms. These are also separated by either addition or subtraction signs. The structure of trinomials makes them versatile elements in algebraic expressions and equations.

In the original exercise, the expression \(3a^2 + 5a - 1\) is identified as a trinomial. This is because it includes three terms: \(3a^2\), \(5a\), and \(-1\). The highest exponent of the variable is 2, making it a quadratic trinomial.

Important aspects about trinomials include:

• Three distinct terms: They always have exactly three separate terms.
• Involves various degrees: The terms can present different power levels of the variable.
• Significance in factoring: Trinomials are common in factoring exercises which often appear in quadratic equations.

Understanding what makes a trinomial helps students to categorize and solve algebraic problems more easily.

Algebraic Expressions

Algebraic expressions form the backbone of algebra, consisting of numbers, variables, and operation symbols. These components are organized to represent particular values or relationships.

In general, algebraic expressions can combine several terms, and the terms may contain a mix of variables and constants. The expressions can range from simple to complex structures.

Crucial points about algebraic expressions:

• Made of terms: A term is a product of numbers and variables.
• Includes operations: Operations are combined within expressions, often causing complexity.
• Varied in structure: From monomials with one term, to polynomials with multiple terms.

In our specific exercise, the algebraic expression \((2a - 4)(3a^2 + 5a - 1)\) is a multiplication of a binomial and a trinomial. Understanding the structure of algebraic expressions can help in solving and simplifying equations across different mathematical problems.