Problem 20 - Intermediate Algebra

Question

What are the terms of the expression? Give the coefficient of each term. See Objective \(1 .\) $$2 y^{4}-y^{3}+6 y+4$$

Step-by-Step Solution

Answer

Terms: \(2y^4\), \(-y^3\), \(6y\), \(4\); Coefficients: \(2\), \(-1\), \(6\), \(4\).

Step 1: Identify the Terms

In the given expression, separate each term by identifying the parts that are added or subtracted. The expression is \(2y^4 - y^3 + 6y + 4\). From this, we identify the terms as \(2y^4\), \(-y^3\), \(6y\), and \(4\).

Step 2: Identify Coefficients of Each Term

For each term, determine the numerical coefficient - this is the number multiplying the variable part. In \(2y^4\), the coefficient is \(2\). In \(-y^3\), the coefficient is \(-1\). In \(6y\), the coefficient is \(6\). Lastly, in the constant term \(4\), the coefficient is \(4\), since it's considered as \(4y^0\).

Key Concepts

Terms in Algebraic Expressions

Algebraic expressions are made up of components called terms. Each term in an expression is a combination of numbers and variables connected together by multiplication or division. Whenever you see an expression like \(2y^4 - y^3 + 6y + 4\), imagine it as a sum of individual parts. These parts are called terms. When you're breaking down an algebraic expression, look for parts separated by plus or minus signs:

• In the expression \(2y^4 - y^3 + 6y + 4\), the plus and minus signs show us that there are four terms: \( 2y^4, -y^3, 6y, \) and \(4\).
• Think of each term as a complete piece of the puzzle, which can include numbers, letters, or a combination of both.

By understanding terms, you break down complex expressions into manageable bites, making it easier to work with algebraic equations.

Coefficients

One important part of any algebraic term is the coefficient. The coefficient is simply the number in front of the variable in a term. It's what tells us how many times we have that particular variable part. Let's break it down using the expression from our exercise:

• In the term \(2y^4\), "2" is the coefficient, showing there are 2 of \(y^4\).
• For \(-y^3\), despite the absence of an explicit number, the coefficient is \(-1\). This tells us there is a \(-1\) multiplier for \(y^3\).
• The coefficient of \(6y\) is "6," indicating six times the variable \(y\).
• In a constant term like \(4\), we can think of the coefficient as "4" times \(1\), or equivalently, \(4y^0\).

Knowing the coefficients helps in understanding the relative size and value of each term within the expression.

Variable Parts

In mathematics, variable parts are the elements of a term that include variables like \(x\), \(y\), or any other letter symbol used to represent numbers in algebraic expressions. The variable part of a term determines how changes in the variables affect the entire term:

• In \(2y^4\), the variable part is \(y^4\), which means this term relies on the power of \(y\).
• In \(-y^3\), the variable is \(y^3\). The expression shows dependence on the cube of \(y\).
• For \(6y\), the variable part is simply \(y\), making it a first-degree polynomial term.
• In a constant, like "4," there is no variable part, or it can be thought of as \(y^0\), since variables to the zero power equal 1.

Variable parts are significant because they shape how terms behave within an expression, especially when you change values of the variables.