Problem 6 - Elementary Algebra

Question

List all of the coefficients and variable parts of the following expressions. $$ -4 a 2 b+5 a b 2-a b+1 $$

Step-by-Step Solution

Answer

Coefficients: -4, 5, -1, 1; Variable parts: \(a^2b\), \(ab^2\), \(ab\).

Step 1: Break Down the Expression

The given expression is \(-4a^2b + 5ab^2 - ab + 1\). Identify each term in the expression: \(-4a^2b\), \(+5ab^2\), \(-ab\), and \(+1\).

Step 2: Identify Coefficients

For each term, identify the coefficient:- In \(-4a^2b\), the coefficient is \(-4\).- In \(+5ab^2\), the coefficient is \(+5\).- In \(-ab\), the coefficient is \(-1\).- In \(+1\), the coefficient is \(+1\) (it is a constant term).

Step 3: Identify Variable Parts

For each term, identify the variable part:- In \(-4a^2b\), the variable part is \(a^2b\).- In \(+5ab^2\), the variable part is \(ab^2\).- In \(-ab\), the variable part is \(ab\).- In \(+1\), there is no variable part since it is a constant.

Key Concepts

Coefficients

In algebraic expressions, coefficients are the numbers that are multiplied with the variable parts. Essentially, they tell us "how much" of a variable term is present in each part of the expression. For example, in the expression \(-4a^2b + 5ab^2 - ab + 1\), each term starts with a coefficient:

• \(-4a^2b\) has a coefficient of \(-4\)
• \(+5ab^2\) has a coefficient of \(+5\)
• \(-ab\) has a coefficient of \(-1\), since \(-1\) is the multiplier for the term \(ab\)
• \(+1\) is a constant term with its coefficient essentially being \(+1\)

These coefficients help us in manipulating the terms of the expression, particularly when performing operations like addition or subtraction of similar terms.

Variable Parts

The variable parts of an algebraic expression are made up of the variables and their exponents attached to each term. They essentially define the "identity" of each term, showing what variables are involved and to what power. Looking at the expression \(-4a^2b + 5ab^2 - ab + 1\), we can inspect the variable parts:

• In \(-4a^2b\), the variable part is \(a^2b\), indicating that it includes \(a\) raised to the power of 2 and \(b\)
• In \(+5ab^2\), the variable part is \(ab^2\), highlighting that \(b\) is squared while \(a\) remains as it is
• In \(-ab\), the variable part is \(ab\), simply incorporating both \(a\) and \(b\) without any exponents attached
• In \(+1\), there is no variable part, as this is the constant term

Understanding variable parts is crucial because they dictate how terms can be combined or simplified in an expression.

Constant Term

A constant term in an expression is a term that does not contain any variables; it is simply a number on its own. It remains constant, meaning its value does not change regardless of the values of any variables in the expression.In our given expression, \(-4a^2b + 5ab^2 - ab + 1\), the constant term is \(+1\). This term stands independently from the other parts of the expression because:

• It does not involve a variable part, distinguishing it from the terms with coefficients tied to variables
• It remains unaffected by any changes made to the variables \(a\) and \(b\)

Recognizing constant terms is important when solving equations or simplifying expressions, as they often act as the base around which simplifications and calculations are centered.