The Difference of Polynomials

Oct 13, 2025 by miodeya.com 11 views

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. In this article, we will explore the difference of polynomials, which is a fundamental concept in algebra. We will examine the given problem and provide a step-by-step solution to determine the correct answer.

Understanding Polynomials

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The variables in a polynomial are often represented by letters such as x, y, or z, while the coefficients are numbers that are multiplied with the variables. Polynomials can be classified into different types based on the degree of the polynomial, which is the highest power of the variable.

The Difference of Polynomials

The difference of polynomials is a fundamental concept in algebra that involves subtracting one polynomial from another. The difference of two polynomials is a new polynomial that results from subtracting the second polynomial from the first. In other words, if we have two polynomials P(x) and Q(x), the difference of P(x) and Q(x) is given by:

P(x) - Q(x)

Given Problem

The given problem is to find the difference of the polynomials:

$\left(m^2 n^2-7\right)-(m n+4)$

To solve this problem, we need to apply the rules of subtraction to the given polynomials.

Step-by-Step Solution

Distribute the negative sign: When subtracting a polynomial, we need to distribute the negative sign to each term in the second polynomial. In this case, we have:

$\left(m^2 n^2-7\right)-\left(m n+4\right)$

Distributing the negative sign, we get:

$m^2 n^2 - m n - 4 - 7$

Combine like terms: Now, we need to combine like terms in the expression. Like terms are terms that have the same variable and exponent. In this case, we have:

$m^2 n^2 - m n - 4 - 7$

Combining the like terms, we get:

$m^2 n^2 - m n - 11$

Conclusion

In conclusion, the difference of the polynomials $\left(m^2 n^2-7\right)-(m n+4)$ is $m^2 n^2 - m n - 11$ . This result is obtained by applying the rules of subtraction to the given polynomials and combining like terms.

Answer

The correct answer is:

C. $m^2 n^2 - m n - 11$

Final Thoughts

In this article, we explored the difference of polynomials, which is a fundamental concept in algebra. We examined the given problem and provided a step-by-step solution to determine the correct answer. The difference of polynomials is an important concept in algebra that has numerous applications in mathematics and science. By understanding the difference of polynomials, we can solve a wide range of problems in algebra and beyond.

Common Mistakes to Avoid

When working with polynomials, it's essential to avoid common mistakes that can lead to incorrect results. Some common mistakes to avoid include:

Not distributing the negative sign: When subtracting a polynomial, it's essential to distribute the negative sign to each term in the second polynomial.
Not combining like terms: Like terms are terms that have the same variable and exponent. It's essential to combine like terms to simplify the expression.
Not following the order of operations: When working with polynomials, it's essential to follow the order of operations (PEMDAS) to ensure that the correct result is obtained.

Real-World Applications

The difference of polynomials has numerous real-world applications in mathematics and science. Some examples include:

Algebraic equations: The difference of polynomials is used to solve algebraic equations, which are equations that involve variables and constants.
Calculus: The difference of polynomials is used in calculus to find the derivative of a function, which is a measure of how the function changes as the input changes.
Engineering: The difference of polynomials is used in engineering to design and analyze complex systems, such as electrical circuits and mechanical systems.

Conclusion

In our previous article, we explored the concept of the difference of polynomials and provided a step-by-step solution to a given problem. In this article, we will answer some frequently asked questions about the difference of polynomials.

Q: What is the difference of polynomials?

A: The difference of polynomials is a fundamental concept in algebra that involves subtracting one polynomial from another. The difference of two polynomials is a new polynomial that results from subtracting the second polynomial from the first.

Q: How do I distribute the negative sign when subtracting polynomials?

A: When subtracting a polynomial, you need to distribute the negative sign to each term in the second polynomial. This means that you need to multiply each term in the second polynomial by -1.

Q: What is the order of operations when working with polynomials?

A: When working with polynomials, you need to follow the order of operations (PEMDAS):

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I combine like terms when subtracting polynomials?

A: Like terms are terms that have the same variable and exponent. When subtracting polynomials, you need to combine like terms by adding or subtracting the coefficients of the like terms.

Q: What are some common mistakes to avoid when working with polynomials?

A: Some common mistakes to avoid when working with polynomials include:

Not distributing the negative sign: When subtracting a polynomial, it's essential to distribute the negative sign to each term in the second polynomial.
Not combining like terms: Like terms are terms that have the same variable and exponent. It's essential to combine like terms to simplify the expression.
Not following the order of operations: When working with polynomials, it's essential to follow the order of operations (PEMDAS) to ensure that the correct result is obtained.

Q: What are some real-world applications of the difference of polynomials?

A: The difference of polynomials has numerous real-world applications in mathematics and science, including:

Algebraic equations: The difference of polynomials is used to solve algebraic equations, which are equations that involve variables and constants.
Calculus: The difference of polynomials is used in calculus to find the derivative of a function, which is a measure of how the function changes as the input changes.
Engineering: The difference of polynomials is used in engineering to design and analyze complex systems, such as electrical circuits and mechanical systems.

Q: How can I practice working with polynomials?

A: There are many ways to practice working with polynomials, including:

Solving problems: Practice solving problems that involve subtracting polynomials.
Using online resources: Use online resources, such as Khan Academy or Mathway, to practice working with polynomials.
Working with a tutor: Consider working with a tutor who can provide personalized instruction and practice.

Conclusion

In conclusion, the difference of polynomials is a fundamental concept in algebra that has numerous applications in mathematics and science. By understanding the difference of polynomials, you can solve a wide range of problems in algebra and beyond. We hope that this Q&A article has provided you with a better understanding of the difference of polynomials and how to apply it in real-world situations.