Understanding the Concept of Like Terms

Oct 13, 2025 by windailys.com 134 views

Which Polynomial Correctly Combines the Like Terms and Expresses the Given Polynomial in Standard Form?

Understanding the Concept of Like Terms

In algebra, like terms are expressions that have the same variable(s) raised to the same power. When combining like terms, we add or subtract their coefficients. For example, in the expression $2x^2 + 3x^2$ , the like terms are $2x^2$ and $3x^2$ . When we combine them, we get $(2+3)x^2 = 5x^2$ .

Given Polynomial and Options

The given polynomial is:

${ 9xy^3 - 4y^4 - 10x^2y^2 + x^3y + 3x^4 + 2x^2y^2 - 9y^4 }$

We need to combine the like terms and express the polynomial in standard form. The options are:

A. ${-13y^4 + 3x^4 - 8x^2y^2 + x^3y}$ B. ${9xy^3 + 3x^4 - 13y^4 - 8x^2y^2 + x^3y}$ C. ${-13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3}$ D. ${-13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3 - 2x^2y^2}$

Step-by-Step Solution

To solve this problem, we need to combine the like terms in the given polynomial.

Step 1: Identify the Like Terms

The like terms in the given polynomial are:

$-4y^4$ and $-9y^4$
$-10x^2y^2$ and $2x^2y^2$
$3x^4$
$x^3y$

Step 2: Combine the Like Terms

Now, we combine the like terms:

$-4y^4 - 9y^4 = -13y^4$
$-10x^2y^2 + 2x^2y^2 = -8x^2y^2$
$3x^4$ remains the same
$x^3y$ remains the same

Step 3: Arrange the Terms in Standard Form

The standard form of a polynomial is arranged in descending order of the powers of the variables. In this case, the variables are $x$ and $y$ . The powers of $x$ are $4$ , $3$ , and $2$ , and the powers of $y$ are $4$ , $3$ , and $2$ . Therefore, the standard form of the polynomial is:

${ -13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3 }$

Step 4: Compare with the Options

Now, we compare the standard form of the polynomial with the options:

A. ${-13y^4 + 3x^4 - 8x^2y^2 + x^3y}$ B. ${9xy^3 + 3x^4 - 13y^4 - 8x^2y^2 + x^3y}$ C. ${-13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3}$ D. ${-13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3 - 2x^2y^2}$

The correct option is:

C. ${-13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3}$

Conclusion

In this article, we learned how to combine like terms and express a polynomial in standard form. We identified the like terms in the given polynomial, combined them, and arranged the terms in standard form. We compared the standard form of the polynomial with the options and found that the correct option is C. ${-13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3}$

Key Takeaways

Like terms are expressions that have the same variable(s) raised to the same power.
When combining like terms, we add or subtract their coefficients.
The standard form of a polynomial is arranged in descending order of the powers of the variables.
To express a polynomial in standard form, we need to combine the like terms and arrange the terms in descending order of the powers of the variables.

Practice Problems

Combine the like terms in the polynomial: $2x^2 + 3x^2 - 4x^2 + 5x^2$
Express the polynomial in standard form: $-3x^4 + 2x^3y - 4x^2y^2 + 5xy^3 - 2y^4$
Combine the like terms in the polynomial: $-2x^2y^2 + 3x^2y^2 - 4x^2y^2 + 5x^2y^2$

Answer Key

$6x^2$
$-3x^4 + 2x^3y - 4x^2y^2 + 5xy^3 - 2y^4$
$2x^2y^2$
Q&A: Combining Like Terms and Expressing Polynomials in Standard Form

Frequently Asked Questions

In this article, we will answer some frequently asked questions about combining like terms and expressing polynomials in standard form.

Q: What are like terms?

A: Like terms are expressions that have the same variable(s) raised to the same power. For example, in the expression $2x^2 + 3x^2$ , the like terms are $2x^2$ and $3x^2$ .

Q: How do I combine like terms?

A: To combine like terms, we add or subtract their coefficients. For example, in the expression $2x^2 + 3x^2$ , we combine the like terms by adding their coefficients: $2 + 3 = 5$ . Therefore, the combined term is $5x^2$ .

Q: What is the standard form of a polynomial?

A: The standard form of a polynomial is arranged in descending order of the powers of the variables. In this case, the variables are $x$ and $y$ . The powers of $x$ are $4$ , $3$ , and $2$ , and the powers of $y$ are $4$ , $3$ , and $2$ . Therefore, the standard form of the polynomial is:

${ -13y^4 + 3x^4 - 8x^2y^2 + x^3y + 9xy^3 }$

Q: How do I express a polynomial in standard form?

A: To express a polynomial in standard form, we need to combine the like terms and arrange the terms in descending order of the powers of the variables.

Q: What are some common mistakes to avoid when combining like terms?

A: Some common mistakes to avoid when combining like terms are:

Not identifying the like terms correctly
Not combining the like terms correctly
Not arranging the terms in descending order of the powers of the variables

Q: How can I practice combining like terms and expressing polynomials in standard form?

A: You can practice combining like terms and expressing polynomials in standard form by working on exercises and problems. You can also use online resources and tools to help you practice.

Q: What are some real-world applications of combining like terms and expressing polynomials in standard form?

A: Combining like terms and expressing polynomials in standard form has many real-world applications, such as:

Solving equations and inequalities
Graphing functions
Finding the roots of a polynomial
Solving systems of equations

Conclusion

In this article, we answered some frequently asked questions about combining like terms and expressing polynomials in standard form. We discussed the concept of like terms, how to combine them, and how to express a polynomial in standard form. We also provided some common mistakes to avoid and some real-world applications of combining like terms and expressing polynomials in standard form.

Key Takeaways

Like terms are expressions that have the same variable(s) raised to the same power.
To combine like terms, we add or subtract their coefficients.
The standard form of a polynomial is arranged in descending order of the powers of the variables.
To express a polynomial in standard form, we need to combine the like terms and arrange the terms in descending order of the powers of the variables.

Practice Problems

Combine the like terms in the polynomial: $2x^2 + 3x^2 - 4x^2 + 5x^2$
Express the polynomial in standard form: $-3x^4 + 2x^3y - 4x^2y^2 + 5xy^3 - 2y^4$
Combine the like terms in the polynomial: $-2x^2y^2 + 3x^2y^2 - 4x^2y^2 + 5x^2y^2$

Answer Key

$6x^2$
$-3x^4 + 2x^3y - 4x^2y^2 + 5xy^3 - 2y^4$
$2x^2y^2$