Add:$ \left(3n^2 + 9n + 3\right) + \left(9n^2 + N + 7\right) $

What is a Polynomial?

A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. In this case, we are dealing with polynomials in the form of $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

The Problem: Adding Two Polynomials

We are given two polynomials to add together:

$$\left(3n^2 + 9n + 3\right) + \left(9n^2 + n + 7\right)$$

Our goal is to simplify this expression by combining like terms.

Step 1: Identify Like Terms

Like terms are terms that have the same variable raised to the same power. In this case, we have two types of like terms:

• Terms with $n^2$
• Terms with $n$
• Constant terms

Step 2: Combine Like Terms

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

• For the terms with $n^2$, we have $3n^2$ and $9n^2$. We can combine these by adding their coefficients: $3n^2 + 9n^2 = 12n^2$.
• For the terms with $n$, we have $9n$ and $n$. We can combine these by adding their coefficients: $9n + n = 10n$.
• For the constant terms, we have $3$ and $7$. We can combine these by adding their coefficients: $3 + 7 = 10$.

Step 3: Simplify the Expression

Now that we have combined the like terms, we can simplify the expression by writing it in the form of a single polynomial.

$$\left(3n^2 + 9n + 3\right) + \left(9n^2 + n + 7\right) = 12n^2 + 10n + 10$$

Conclusion

Adding polynomials involves identifying like terms and combining them by adding or subtracting their coefficients. By following these steps, we can simplify complex expressions and write them in a more concise form.

Tips and Tricks

• When adding polynomials, make sure to identify all like terms, including constant terms.
• Use the distributive property to expand expressions and make it easier to identify like terms.
• Combine like terms by adding or subtracting their coefficients.

Real-World Applications

Adding polynomials has many real-world applications, including:

• Algebraic geometry: Polynomials are used to define curves and surfaces in algebraic geometry.
• Computer science: Polynomials are used in computer science to represent and manipulate data.
• Engineering: Polynomials are used in engineering to model and analyze complex systems.

Common Mistakes

• Failing to identify all like terms, including constant terms.
• Not using the distributive property to expand expressions.
• Not combining like terms correctly.

Practice Problems

1. Add the following polynomials: $\left(2x^2 + 3x + 1\right) + \left(x^2 + 2x + 4\right)$
2. Add the following polynomials: $\left(3y^2 + 2y + 5\right) + \left(y^2 + 3y + 2\right)$

Answer Key

1. $\left(2x^2 + 3x + 1\right) + \left(x^2 + 2x + 4\right) = 3x^2 + 5x + 5$
2. $\left(3y^2 + 2y + 5\right) + \left(y^2 + 3y + 2\right) = 4y^2 + 5y + 7$
   Q&A: Adding Polynomials ==========================

Q: What is the first step in adding polynomials?

A: The first step in adding polynomials is to identify like terms. Like terms are terms that have the same variable raised to the same power.

Q: How do I identify like terms?

A: To identify like terms, look for terms that have the same variable raised to the same power. For example, in the polynomial $2x^2 + 3x + 1$, the terms $2x^2$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: What is the next step after identifying like terms?

A: After identifying like terms, the next step is to combine them by adding or subtracting their coefficients. For example, if we have the terms $2x^2$ and $3x^2$, we can combine them by adding their coefficients: $2x^2 + 3x^2 = 5x^2$.

Q: How do I combine like terms?

A: To combine like terms, add or subtract their coefficients. For example, if we have the terms $2x$ and $3x$, we can combine them by adding their coefficients: $2x + 3x = 5x$.

Q: What is the final step in adding polynomials?

A: The final step in adding polynomials is to simplify the expression by writing it in the form of a single polynomial. For example, if we have the expression $2x^2 + 3x + 1 + x^2 + 2x + 4$, we can simplify it by combining like terms: $2x^2 + x^2 = 3x^2$, $3x + 2x = 5x$, and $1 + 4 = 5$. The simplified expression is $3x^2 + 5x + 5$.

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

A: Some common mistakes to avoid when adding polynomials include:

• Failing to identify all like terms, including constant terms.
• Not using the distributive property to expand expressions.
• Not combining like terms correctly.

Q: How do I use the distributive property to expand expressions?

A: To use the distributive property to expand expressions, multiply each term in the expression by the coefficient of the term. For example, if we have the expression $2(x^2 + 3x + 1)$, we can expand it by multiplying each term by 2: $2x^2 + 6x + 2$.

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

A: Some real-world applications of adding polynomials include:

• Algebraic geometry: Polynomials are used to define curves and surfaces in algebraic geometry.
• Computer science: Polynomials are used in computer science to represent and manipulate data.
• Engineering: Polynomials are used in engineering to model and analyze complex systems.

Q: How do I practice adding polynomials?

A: To practice adding polynomials, try the following exercises:

• Add the following polynomials: $\left(2x^2 + 3x + 1\right) + \left(x^2 + 2x + 4\right)$
• Add the following polynomials: $\left(3y^2 + 2y + 5\right) + \left(y^2 + 3y + 2\right)$

Q: What are some tips for adding polynomials?

A: Some tips for adding polynomials include:

• Make sure to identify all like terms, including constant terms.
• Use the distributive property to expand expressions.
• Combine like terms correctly.

Q: How do I check my work when adding polynomials?

A: To check your work when adding polynomials, try the following:

• Simplify the expression by combining like terms.
• Check that the expression is in the form of a single polynomial.
• Verify that the expression is correct by plugging in values for the variable.