Problem 105 - Algebra A Combined Function

Question

Construct a binomial whose greatest common factor is \(5 a^{3}\). (Hint: Multiply \(5 a^{3}\) by a binomial whose terms contain no common factor other than \(\left.1: 5 a^{3}(\square+\square) .\right)\)

Step-by-Step Solution

Answer

The binomial is \(5a^3 + 10a^3\).

Step 1: Understand the Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) of a binomial is the largest factor that divides all terms of the binomial. Here, we need a binomial with the GCF of \(5a^3\). This means that each term in the binomial should be divisible by \(5a^3\).

Step 2: Choose Terms for the Binomial

To construct the binomial, choose two terms such that their only common factor is \(1\) after factoring out \(5a^3\). For simplicity, let's choose \(b\) and \(c\) such that the binomial is \((b + c)\).

Step 3: Multiply Each Term by the GCF

Multiply \(5a^3\) by each term separately to construct the binomial. This gives us \(5a^3(b) + 5a^3(c)\).

Step 4: Construct the Binomial

The binomial with the GCF \(5a^3\) is \(5a^3(b + c)\). For simplicity, we can let \(b = 1\) and \(c = 2\) making the binomial \(5a^3(1 + 2) = 5a^3 \cdot 1 + 5a^3 \cdot 2 = 5a^3 + 10a^3\).

Key Concepts

Binomial

A binomial is an algebraic expression that consists of two distinct terms. It is one of the simplest and most basic forms of polynomial expressions, which typically consist of multiple terms. In a binomial, these two terms are usually joined by a plus or minus sign. For example, in the expression

• \(a + b\)
• \(x - y\)

both are examples of binomials, indicating two separate components that behave as a single unit when operations are applied.
When constructing a binomial, especially with specific conditions like having a given Greatest Common Factor (GCF), it's crucial to carefully select the terms such that they satisfy the given conditions while maintaining the form of a binomial.

Factoring

Factoring involves breaking down an expression into smaller, simpler components called factors. This is like splitting a large number or expression into products of simpler or smaller numbers or expressions that, when multiplied together, yield the original expression.
When we talk about factoring in the context of algebraic expressions, we usually seek to express the given number as a product of its factors. For instance,

• The factorization of \( 12a^3b^2 \) might yield products like \( 3 \times 4 \times a \times a \times a \times b \times b \).

In the given exercise, the idea is to factor out the GCF of a binomial, which is \( 5a^3 \). This means that we identify and separate this term from each part of the binomial, ensuring that it remains a factor common to both terms in the resulting expression.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and operational symbols. These expressions can be simple like

• \(x+7\)
• \(3y^2\)

or more complex involving multiple terms like

• \(2x^2 - 3x + 5\).

Such expressions are fundamental in algebra as they form the basis of equations and functions.
An important component of algebraic expressions is understanding how each part functions together. Variables represent unknowns or changing values, and constants provide fixed values. Operations define the relationships between these components. In the exercise we addressed, we are essentially asked to form an algebraic expression—specifically, a binomial—by controlling the placement and multiplication of factors to achieve a specific form with a designated Greatest Common Factor (GCF).