What Is The Sum Of The Polynomials?${ \begin{array}{r} 17m - 12n - 1 \ , 4 - 13m - 12n \ \hline \end{array} }$A. ${ 4m + 3\$} B. ${ 4m - 24n + 3\$} C. ${ 30m - 5\$} D. ${ 30m - 24\pi - 5\$}
Introduction
In algebra, polynomials are expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When we add two polynomials, we combine like terms to simplify the expression. In this article, we will explore how to find the sum of two polynomials and provide a step-by-step solution to a given problem.
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. For example, the expression 2x + 3y - 4 is a polynomial.
Adding Polynomials
When we add two polynomials, we combine like terms to simplify the expression. Like terms are terms that have the same variable raised to the same power. For example, 2x and 3x are like terms, while 2x and 3y are not.
To add two polynomials, we follow these steps:
- Combine like terms by adding or subtracting the coefficients of the like terms.
- Simplify the expression by combining the like terms.
Step-by-Step Solution
Let's consider the given problem:
{ \begin{array}{r} 17m - 12n - 1 \\ + \, 4 - 13m - 12n \\ \hline \end{array} \}
To find the sum of the polynomials, we will follow the steps outlined above.
Step 1: Combine Like Terms
The first polynomial is 17m - 12n - 1, and the second polynomial is 4 - 13m - 12n. We will combine like terms by adding or subtracting the coefficients of the like terms.
- The like terms in the first polynomial are 17m and -13m.
- The like terms in the second polynomial are -12n and -12n.
- The constant terms are -1 and 4.
Step 2: Simplify the Expression
Now that we have combined the like terms, we will simplify the expression by combining the like terms.
- The like terms 17m and -13m combine to give 4m.
- The like terms -12n and -12n combine to give -24n.
- The constant terms -1 and 4 combine to give 3.
Therefore, the sum of the polynomials is 4m - 24n + 3.
Conclusion
In this article, we explored how to find the sum of two polynomials. We followed the steps outlined above to combine like terms and simplify the expression. The sum of the given polynomials is 4m - 24n + 3.
Answer
The correct answer is B. .
Discussion
This problem requires the application of algebraic concepts, specifically the addition of polynomials. The student must be able to identify like terms and combine them to simplify the expression. This problem is relevant to the topic of algebra and is a good example of how to apply algebraic concepts to solve problems.
Related Topics
- Adding and subtracting polynomials
- Combining like terms
- Simplifying expressions
- Algebraic concepts
Glossary
- Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
- Like terms: Terms that have the same variable raised to the same power.
- Coefficient: A number that is multiplied with a variable.
- Variable: A letter or symbol that represents a value that can change.
References
- Algebraic Concepts
- Adding and Subtracting Polynomials
- Combining Like Terms
Q&A: Adding Polynomials ==========================
Introduction
In our previous article, we explored how to find the sum of two polynomials. In this article, we will answer some frequently asked questions about adding polynomials.
Q: What are like terms?
A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 3x are like terms, while 2x and 3y are not.
Q: How do I combine like terms?
A: To combine like terms, you add or subtract the coefficients of the like terms. For example, 2x + 3x = 5x, and 2x - 3x = -x.
Q: What is the difference between a polynomial and an expression?
A: A polynomial is a type of expression that consists of variables and coefficients combined using only addition, subtraction, and multiplication. An expression is a general term that can include variables, coefficients, and constants.
Q: Can I add polynomials with different variables?
A: Yes, you can add polynomials with different variables. For example, 2x + 3y and 4x + 5y can be added by combining like terms.
Q: How do I simplify an expression after adding polynomials?
A: To simplify an expression after adding polynomials, you combine like terms and remove any unnecessary parentheses.
Q: What is the order of operations when adding polynomials?
A: The order of operations when adding polynomials is:
- Combine like terms.
- Simplify the expression.
Q: Can I subtract polynomials?
A: Yes, you can subtract polynomials by changing the signs of the terms in the second polynomial and then adding the two polynomials.
Q: How do I multiply polynomials?
A: To multiply polynomials, you multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
Q: What is the difference between multiplying and adding polynomials?
A: Multiplying polynomials involves multiplying each term in the first polynomial by each term in the second polynomial, while adding polynomials involves combining like terms.
Q: Can I add polynomials with negative coefficients?
A: Yes, you can add polynomials with negative coefficients. For example, -2x + 3y and 4x - 5y can be added by combining like terms.
Q: How do I handle fractions when adding polynomials?
A: When adding polynomials with fractions, you can multiply each term by the least common multiple (LCM) of the denominators to eliminate the fractions.
Conclusion
In this article, we answered some frequently asked questions about adding polynomials. We covered topics such as like terms, combining like terms, and simplifying expressions. We also discussed the order of operations when adding polynomials and how to handle fractions.
Glossary
- Polynomial: An expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication.
- Like terms: Terms that have the same variable raised to the same power.
- Coefficient: A number that is multiplied with a variable.
- Variable: A letter or symbol that represents a value that can change.
- Expression: A general term that can include variables, coefficients, and constants.
- Least common multiple (LCM): The smallest multiple that two or more numbers have in common.