Combining Like Terms in the Given Polynomial

Oct 13, 2025 by magicalceylontours.com 92 views

Understanding the Basics of Polynomial Combination

When dealing with polynomials, combining like terms is a crucial step in simplifying the expression. This process involves adding or subtracting terms that have the same variable and exponent. In this article, we will explore the terms that can be combined in the given polynomial $$x^4 + 6x^2 - 5x + 2x^3 - 9x^2$$.

What are Like Terms?

Like terms are terms that have the same variable and exponent. For example, in the polynomial $$x^2 + 3x^2 + 2x^3$$, the terms $$x^2$$ and $$3x^2$$ are like terms because they have the same variable (x) and exponent (2). Similarly, the terms $$2x^3$$ and any other term with the variable x and exponent 3 are also like terms.

Combining Like Terms in the Given Polynomial

To combine like terms in the given polynomial $$x^4 + 6x^2 - 5x + 2x^3 - 9x^2$$, we need to identify the like terms and add or subtract their coefficients.

Step 1: Identify the Like Terms

The like terms in the given polynomial are:

$x^4$$ and no other term with the same exponent$
$6x^2$$ and $$-9x^2$$, which have the same variable (x) and exponent (2)$
$-5x$$ and $$2x^3$$, which do not have the same variable or exponent$

Step 2: Combine the Like Terms

Now that we have identified the like terms, we can combine them by adding or subtracting their coefficients.

The term $$x^4$$ remains unchanged because there is no other term with the same exponent.
The like terms $$6x^2$$ and $$-9x^2$$ can be combined by adding their coefficients: $$6x^2 - 9x^2 = -3x^2$$
The terms $$-5x$$ and $$2x^3$$ do not have the same variable or exponent, so they cannot be combined.

Step 3: Simplify the Polynomial

After combining the like terms, the polynomial becomes:

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

This is the simplified form of the given polynomial.

Conclusion

Combining like terms is an essential step in simplifying polynomials. By identifying the like terms and adding or subtracting their coefficients, we can simplify the polynomial and make it easier to work with. In this article, we have explored the terms that can be combined in the given polynomial $$x^4 + 6x^2 - 5x + 2x^3 - 9x^2$$ and simplified it to $$x^4 - 3x^2 + 2x^3 - 5x$$.

Common Mistakes to Avoid

When combining like terms, it's essential to avoid common mistakes such as:

Not identifying like terms: Make sure to identify all the like terms in the polynomial before combining them.
Incorrectly combining like terms: Double-check that you are combining like terms correctly by adding or subtracting their coefficients.
Forgetting to simplify the polynomial: After combining like terms, make sure to simplify the polynomial by combining any remaining like terms.

Real-World Applications

Combining like terms has numerous real-world applications in various fields such as:

Physics: When solving problems involving motion, combining like terms can help simplify the equations and make it easier to find the solution.
Engineering: In engineering, combining like terms can help simplify complex equations and make it easier to design and optimize systems.
Computer Science: In computer science, combining like terms can help simplify algorithms and make it easier to write efficient code.

Final Thoughts

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about combining like terms in polynomials.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, in the polynomial $$x^2 + 3x^2 + 2x^3$$, the terms $$x^2$$ and $$3x^2$$ are like terms because they have the same variable (x) and exponent (2).

Q: How do I identify like terms in a polynomial?

A: To identify like terms in a polynomial, you need to look for terms that have the same variable and exponent. You can do this by:

Looking for terms with the same variable: Identify the variable in each term and look for terms that have the same variable.
Looking for terms with the same exponent: Identify the exponent in each term and look for terms that have the same exponent.
Combining terms with the same variable and exponent: Once you have identified the like terms, you can combine them by adding or subtracting their coefficients.

Q: How do I combine like terms in a polynomial?

A: To combine like terms in a polynomial, you need to add or subtract their coefficients. Here are the steps:

Identify the like terms: Identify the like terms in the polynomial.
Add or subtract the coefficients: Add or subtract the coefficients of the like terms.
Simplify the polynomial: Simplify the polynomial by combining any remaining like terms.

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

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

Not identifying like terms: Make sure to identify all the like terms in the polynomial before combining them.
Incorrectly combining like terms: Double-check that you are combining like terms correctly by adding or subtracting their coefficients.
Forgetting to simplify the polynomial: After combining like terms, make sure to simplify the polynomial by combining any remaining like terms.

Q: How do I know if a term is a like term or not?

A: To determine if a term is a like term or not, you need to look at the variable and exponent of the term. If the term has the same variable and exponent as another term, it is a like term.

Q: Can I combine like terms in a polynomial with multiple variables?

A: Yes, you can combine like terms in a polynomial with multiple variables. However, you need to make sure that the terms have the same variable and exponent.

Q: How do I simplify a polynomial with multiple like terms?

A: To simplify a polynomial with multiple like terms, you need to combine the like terms by adding or subtracting their coefficients. Here are the steps:

Identify the like terms: Identify the like terms in the polynomial.
Add or subtract the coefficients: Add or subtract the coefficients of the like terms.
Simplify the polynomial: Simplify the polynomial by combining any remaining like terms.

Q: Can I use a calculator to combine like terms in a polynomial?

A: Yes, you can use a calculator to combine like terms in a polynomial. However, it's always a good idea to double-check your work by hand to make sure that you are combining like terms correctly.

Conclusion

Combining like terms is an essential step in simplifying polynomials. By identifying the like terms and adding or subtracting their coefficients, we can simplify the polynomial and make it easier to work with. In this article, we have addressed some of the most frequently asked questions about combining like terms in polynomials. We hope that this article has provided you with a better understanding of combining like terms and how it can be applied in real-world scenarios.