Classification of Expressions by Degree and Number of Terms

Oct 13, 2025 by windailys.com 3 views

In this article, we will simplify or expand each given expression and then classify it by its degree and number of terms. We will use algebraic techniques to simplify the expressions and determine their degree and number of terms.

Expression 1: $2x^3(7x^2 + 3x + 1) - 4x^4$

To simplify this expression, we will use the distributive property to expand the product of $2x^3$ and $(7x^2 + 3x + 1)$ .

2x^3(7x^2 + 3x + 1) = 14x^5 + 6x^4 + 2x^3

Now, we can rewrite the original expression as:

14x^5 + 6x^4 + 2x^3 - 4x^4

Combining like terms, we get:

14x^5 + (6x^4 - 4x^4) + 2x^3

Simplifying further, we get:

14x^5 + 2x^4 + 2x^3

The degree of this expression is 5, and the number of terms is 3.

Expression 2: $(6x - 2x^4 + 5x^6) + (7x^4 - 3x^6 + 9)$

To simplify this expression, we will combine like terms.

(6x - 2x^4 + 5x^6) + (7x^4 - 3x^6 + 9) = 6x - 2x^4 + 5x^6 + 7x^4 - 3x^6 + 9

Combining like terms, we get:

6x + (7x^4 - 2x^4) + (5x^6 - 3x^6) + 9

Simplifying further, we get:

6x + 5x^4 + 2x^6 + 9

The degree of this expression is 6, and the number of terms is 4.

Expression 3: $(x^2 + 9)(x^2 - 9)$

To simplify this expression, we will use the difference of squares formula:

(x^2 + 9)(x^2 - 9) = (x^2)^2 - (9)^2

Simplifying further, we get:

x^4 - 81

The degree of this expression is 4, and the number of terms is 1.

Classification of Expressions by Degree and Number of Terms

Expression	Degree	Number of Terms
$2x^3(7x^2 + 3x + 1) - 4x^4$	5	3
$(6x - 2x^4 + 5x^6) + (7x^4 - 3x^6 + 9)$	6	4
$(x^2 + 9)(x^2 - 9)$	4	1

In conclusion, we have simplified or expanded each given expression and classified it by its degree and number of terms. We have used algebraic techniques to simplify the expressions and determine their degree and number of terms.

Degree of an Expression

The degree of an expression is the highest power of the variable in the expression. For example, in the expression $x^3 + 2x^2 + 3x + 1$ , the degree is 3 because the highest power of the variable $x$ is 3.

Number of Terms in an Expression

The number of terms in an expression is the number of individual terms in the expression. For example, in the expression $x^3 + 2x^2 + 3x + 1$ , the number of terms is 4 because there are four individual terms in the expression.

Importance of Degree and Number of Terms

The degree and number of terms in an expression are important because they can affect the behavior of the expression. For example, an expression with a high degree may have a more complex behavior than an expression with a low degree. Similarly, an expression with a large number of terms may be more difficult to work with than an expression with a small number of terms.

Conclusion

In this article, we will answer some frequently asked questions about simplifying and classifying algebraic expressions.

Q: What is the difference between simplifying and classifying an algebraic expression?

A: Simplifying an algebraic expression involves combining like terms and removing any unnecessary parentheses or other notation. Classifying an algebraic expression involves determining its degree and number of terms.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you can use the following steps:

Combine like terms by adding or subtracting the coefficients of the same variables.
Remove any unnecessary parentheses or other notation.
Simplify any fractions or other expressions within the expression.

Q: How do I classify an algebraic expression?

A: To classify an algebraic expression, you can use the following steps:

Determine the degree of the expression by finding the highest power of the variable.
Determine the number of terms in the expression by counting the individual terms.

Q: What is the degree of an algebraic expression?

A: The degree of an algebraic expression is the highest power of the variable in the expression. For example, in the expression $x^3 + 2x^2 + 3x + 1$ , the degree is 3 because the highest power of the variable $x$ is 3.

Q: What is the number of terms in an algebraic expression?

A: The number of terms in an algebraic expression is the number of individual terms in the expression. For example, in the expression $x^3 + 2x^2 + 3x + 1$ , the number of terms is 4 because there are four individual terms in the expression.

Q: Why is it important to simplify and classify algebraic expressions?

A: Simplifying and classifying algebraic expressions is important because it can help you:

Understand the behavior of the expression
Solve equations and inequalities
Graph functions
Perform other mathematical operations

Q: Can you give an example of a simplified and classified algebraic expression?

A: Here is an example of a simplified and classified algebraic expression:

Expression: $2x^3(7x^2 + 3x + 1) - 4x^4$

Simplified: $14x^5 + 2x^4 + 2x^3$

Classified: Degree: 5, Number of Terms: 3

Q: How do I determine the degree of a polynomial expression?

A: To determine the degree of a polynomial expression, you can use the following steps:

Identify the highest power of the variable in the expression.
Determine the degree of the expression by finding the highest power of the variable.

Q: How do I determine the number of terms in a polynomial expression?

A: To determine the number of terms in a polynomial expression, you can use the following steps:

Count the individual terms in the expression.
Determine the number of terms by counting the individual terms.

Q: Can you give an example of a polynomial expression with a high degree?

A: Here is an example of a polynomial expression with a high degree:

Expression: $x^5 + 2x^4 + 3x^3 + 4x^2 + 5x + 6$

Degree: 5

Number of Terms: 6

Q: Can you give an example of a polynomial expression with a low degree?

A: Here is an example of a polynomial expression with a low degree:

Expression: $x + 2$

Degree: 1

Number of Terms: 2

Conclusion

In this article, we have answered some frequently asked questions about simplifying and classifying algebraic expressions. We have discussed the importance of simplifying and classifying algebraic expressions and provided examples of simplified and classified expressions. We have also discussed how to determine the degree and number of terms in a polynomial expression.