Add The Polynomials:$\[ \begin{array}{r} 3x^2 - 5x + 1 \\ +\quad 2x^2 + 9x - 6 \\ \hline \end{array} \\]Choose The Correct Answer:A. \[$5x^2 - 4x + 5\$\] B. \[$5x^2 + 14x - 5\$\] C. \[$5x^2 + 4x + 7\$\] D.

Introduction

Polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, and multiplication. When adding polynomials, we combine like terms, which are terms that have the same variable and exponent. In this article, we will explore how to add polynomials, using the given example of adding two quadratic polynomials.

What are Polynomials?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. Polynomials can be classified based on the degree of the polynomial, which is the highest power of the variable. For example, a quadratic polynomial has a degree of 2, while a linear polynomial has a degree of 1.

Adding Polynomials: A Step-by-Step Guide

When adding polynomials, we combine like terms, which are terms that have the same variable and exponent. To add polynomials, follow these steps:

1. Identify like terms: Identify the terms in each polynomial that have the same variable and exponent.
2. Combine like terms: Combine the like terms by adding their coefficients.
3. Write the result: Write the resulting polynomial by combining the like terms.

Example: Adding Two Quadratic Polynomials

Let's consider the example of adding two quadratic polynomials:

{ \begin{array}{r} 3x^2 - 5x + 1 \\ +\quad 2x^2 + 9x - 6 \\ \hline \end{array} \}

To add these polynomials, we need to combine like terms. The like terms in this example are the terms with the same variable and exponent.

• The like terms with the variable $x^2$ are $3x^2$ and $2x^2$.
• The like terms with the variable $x$ are $-5x$ and $9x$.
• The like terms with the constant term are $1$ and $-6$.

Now, let's combine the like terms:

• The sum of the like terms with the variable $x^2$ is $3x^2 + 2x^2 = 5x^2$.
• The sum of the like terms with the variable $x$ is $-5x + 9x = 4x$.
• The sum of the like terms with the constant term is $1 - 6 = -5$.

Therefore, the resulting polynomial is:

{ 5x^2 + 4x - 5 \}

Conclusion

Adding polynomials involves combining like terms, which are terms that have the same variable and exponent. By following the steps outlined in this article, you can add polynomials with ease. Remember to identify like terms, combine them by adding their coefficients, and write the resulting polynomial.

Answer

The correct answer is:

{ \boxed{5x^2 + 4x - 5} \}

This is the resulting polynomial after adding the two quadratic polynomials:

{ \begin{array}{r} 3x^2 - 5x + 1 \\ +\quad 2x^2 + 9x - 6 \\ \hline \end{array} \}

Discussion

What are some common mistakes to avoid when adding polynomials? How do you handle polynomials with negative coefficients? Share your thoughts and experiences in the comments below!

Related Topics

• Multiplying Polynomials
• Dividing Polynomials
• Simplifying Polynomials

Further Reading

• Polynomial Addition
• Algebraic Expressions
• Mathematics
  Adding Polynomials: A Q&A Guide =====================================

Introduction

Adding polynomials is a fundamental concept in algebra, and it's essential to understand how to add polynomials correctly. In this article, we'll answer some frequently asked questions about adding polynomials, providing you with a deeper understanding of this concept.

Q&A

Q: What are like terms in polynomials?

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

Q: How do I combine like terms in polynomials?

A: To combine like terms, add their coefficients. For example, in the polynomial $3x^2 + 2x^2$, the coefficients are $3$ and $2$. Adding these coefficients gives us $5x^2$.

Q: What if I have a polynomial with negative coefficients?

A: When adding polynomials with negative coefficients, remember that a negative coefficient is equivalent to adding the opposite of the term. For example, in the polynomial $-3x^2 + 2x^2$, the coefficient $-3$ is equivalent to adding $-3x^2$, which is the opposite of $3x^2$.

Q: Can I add polynomials with different variables?

A: No, you cannot add polynomials with different variables. For example, you cannot add the polynomial $3x^2 + 2x^2$ to the polynomial $4y^2 + 3y^2$ because they have different variables $x$ and $y$.

Q: How do I add polynomials with multiple terms?

A: To add polynomials with multiple terms, follow the same steps as adding polynomials with a single term. Identify like terms, combine them by adding their coefficients, and write the resulting polynomial.

Q: Can I add polynomials with fractions?

A: Yes, you can add polynomials with fractions. When adding fractions, remember to find a common denominator and add the numerators.

Q: What if I have a polynomial with a variable raised to a power?

A: When adding polynomials with a variable raised to a power, remember to combine like terms with the same variable and exponent.

Q: Can I add polynomials with exponents?

A: Yes, you can add polynomials with exponents. When adding exponents, remember to combine like terms with the same variable and exponent.

Q: How do I simplify the resulting polynomial after adding polynomials?

A: To simplify the resulting polynomial, combine like terms and write the polynomial in the simplest form possible.

Conclusion

Adding polynomials is a fundamental concept in algebra, and it's essential to understand how to add polynomials correctly. By following the steps outlined in this article, you can add polynomials with ease. Remember to identify like terms, combine them by adding their coefficients, and write the resulting polynomial.

Related Topics

• Multiplying Polynomials
• Dividing Polynomials
• Simplifying Polynomials

Further Reading

• Polynomial Addition
• Algebraic Expressions
• Mathematics

Practice Problems

• Add the polynomials $2x^2 + 3x + 1$ and $x^2 + 2x - 1$.
• Add the polynomials $3x^2 - 2x + 1$ and $2x^2 + 3x - 1$.
• Add the polynomials $x^2 + 2x - 1$ and $2x^2 + 3x + 1$.

Answer Key

• $3x^2 + 5x + 0$
• $5x^2 + 5x + 0$
• $3x^2 + 5x + 0$