Combine The Polynomials:${ \begin{array}{c} 2x^2 + 4x + 3 \ (-4x^2 - 10x + 9) \ \hline \end{array} }$

Introduction

Polynomials are algebraic expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. When combining polynomials, we need to add or subtract like terms, which are terms with the same variable and exponent. In this article, we will discuss how to combine polynomials using the given example.

The Problem

The problem is to combine the polynomials:

$$\begin{array}{c} 2x^2 + 4x + 3 \\ + (-4x^2 - 10x + 9) \\ \hline \end{array}$$

Step 1: Identify Like Terms

To combine polynomials, we need to identify like terms. Like terms are terms with the same variable and exponent. In this example, we have two polynomials:

$$2x^2 + 4x + 3$$

and

$$-4x^2 - 10x + 9$$

We can see that the first polynomial has a term $2x^2$ and the second polynomial has a term $-4x^2$. These two terms are like terms because they have the same variable ($x$) and exponent ($2$).

Step 2: Combine Like Terms

Now that we have identified like terms, we can combine them. To combine like terms, we add or subtract their coefficients. In this case, we have:

$$2x^2 + (-4x^2)$$

The coefficient of the first term is $2$ and the coefficient of the second term is $-4$. When we add these two terms, we get:

$$-2x^2$$

So, the combined term is $-2x^2$.

Step 3: Combine Remaining Terms

Now that we have combined the like terms, we can combine the remaining terms. The remaining terms are:

$$4x + (-10x)$$

and

$$3 + 9$$

We can see that the first term has a coefficient of $4$ and the second term has a coefficient of $-10$. When we add these two terms, we get:

$$-6x$$

The third term is a constant term, which is $3 + 9 = 12$.

Step 4: Write the Final Answer

Now that we have combined all the terms, we can write the final answer. The final answer is:

$$-2x^2 - 6x + 12$$

Conclusion

Combining polynomials is an important skill in algebra. By following the steps outlined in this article, we can combine polynomials using addition and subtraction. Remember to identify like terms and combine them by adding or subtracting their coefficients. With practice, you will become proficient in combining polynomials.

Example 1: Combining Polynomials with Three Terms

Let's consider another example of combining polynomials:

$$3x^2 + 2x - 4$$

$$+ (-2x^2 - 3x + 5)$$

To combine these polynomials, we need to identify like terms and combine them. The like terms are:

$$3x^2 + (-2x^2)$$

$$2x + (-3x)$$

$$-4 + 5$$

We can see that the first term has a coefficient of $3$ and the second term has a coefficient of $-2$. When we add these two terms, we get:

$$x^2$$

The second term has a coefficient of $2$ and the second term has a coefficient of $-3$. When we add these two terms, we get:

$$-x$$

The third term is a constant term, which is $-4 + 5 = 1$.

So, the final answer is:

$$x^2 - x + 1$$

Example 2: Combining Polynomials with Four Terms

Let's consider another example of combining polynomials:

$$2x^2 + 3x - 2$$

$$+ (-4x^2 - 2x + 3)$$

To combine these polynomials, we need to identify like terms and combine them. The like terms are:

$$2x^2 + (-4x^2)$$

$$3x + (-2x)$$

$$-2 + 3$$

We can see that the first term has a coefficient of $2$ and the second term has a coefficient of $-4$. When we add these two terms, we get:

$$-2x^2$$

The second term has a coefficient of $3$ and the second term has a coefficient of $-2$. When we add these two terms, we get:

$$x$$

The third term is a constant term, which is $-2 + 3 = 1$.

So, the final answer is:

$$-2x^2 + x + 1$$

Tips and Tricks

Here are some tips and tricks to help you combine polynomials:

• Identify like terms: Like terms are terms with the same variable and exponent. Identify like terms and combine them by adding or subtracting their coefficients.
• Combine like terms: Combine like terms by adding or subtracting their coefficients.
• Combine remaining terms: Combine the remaining terms by adding or subtracting their coefficients.
• Write the final answer: Write the final answer by combining all the terms.

By following these tips and tricks, you will become proficient in combining polynomials.

Conclusion

Introduction

Combining polynomials is an important skill in algebra. In our previous article, we discussed how to combine polynomials using addition and subtraction. In this article, we will answer some frequently asked questions about combining polynomials.

Q: What are like terms?

A: Like terms are terms with the same variable and exponent. For example, $2x^2$ and $-4x^2$ are like terms because they have the same variable ($x$) and exponent ($2$).

Q: How do I identify like terms?

A: To identify like terms, you need to look for terms with the same variable and exponent. For example, in the polynomial $2x^2 + 3x - 4$, the like terms are $2x^2$ and $-4$ (because they have the same variable and exponent).

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract their coefficients. For example, in the polynomial $2x^2 + (-4x^2)$, the coefficients are $2$ and $-4$. When you add these two terms, you get $-2x^2$.

Q: What if I have a term with a negative coefficient?

A: If you have a term with a negative coefficient, you need to change the sign of the term when you combine it with another term. For example, in the polynomial $2x^2 + (-4x^2)$, the second term has a negative coefficient. When you combine these two terms, you get $-2x^2$.

Q: Can I combine polynomials with different variables?

A: No, you cannot combine polynomials with different variables. For example, you cannot combine the polynomials $2x^2 + 3x - 4$ and $2y^2 + 3y - 4$ because they have different variables ($x$ and $y$).

Q: Can I combine polynomials with different exponents?

A: No, you cannot combine polynomials with different exponents. For example, you cannot combine the polynomials $2x^2 + 3x - 4$ and $2x^3 + 3x - 4$ because they have different exponents ($2$ and $3$).

Q: How do I write the final answer?

A: To write the final answer, you need to combine all the terms and simplify the expression. For example, in the polynomial $2x^2 + 3x - 4 + (-4x^2 - 2x + 3)$, the final answer is $-2x^2 + x + 1$.

Q: What if I have a polynomial with a zero coefficient?

A: If you have a polynomial with a zero coefficient, you can ignore it when you combine the polynomials. For example, in the polynomial $2x^2 + 3x - 4 + 0x^2 + 0x - 4$, the final answer is $2x^2 + 3x - 8$.

Q: Can I combine polynomials with fractions?

A: Yes, you can combine polynomials with fractions. For example, in the polynomial $\frac{2}{3}x^2 + \frac{3}{4}x - \frac{4}{5}$, you can combine the terms by finding a common denominator.

Conclusion

Combining polynomials is an important skill in algebra. By following the steps outlined in this article, you can combine polynomials using addition and subtraction. Remember to identify like terms and combine them by adding or subtracting their coefficients. With practice, you will become proficient in combining polynomials.

Example 1: Combining Polynomials with Fractions

Let's consider an example of combining polynomials with fractions:

$$\frac{2}{3}x^2 + \frac{3}{4}x - \frac{4}{5}$$

$$+ (-\frac{2}{3}x^2 - \frac{3}{4}x + \frac{4}{5})$$

To combine these polynomials, we need to find a common denominator. The common denominator is $60$. We can rewrite the polynomials as:

$$\frac{40}{60}x^2 + \frac{45}{60}x - \frac{48}{60}$$

$$+ (-\frac{40}{60}x^2 - \frac{45}{60}x + \frac{48}{60})$$

We can see that the like terms are:

$$\frac{40}{60}x^2 + (-\frac{40}{60}x^2)$$

$$\frac{45}{60}x + (-\frac{45}{60}x)$$

$$-\frac{48}{60} + \frac{48}{60}$$

We can combine these terms by adding or subtracting their coefficients. The final answer is:

$$0x^2 + 0x + 0$$

Example 2: Combining Polynomials with Zero Coefficient

Let's consider an example of combining polynomials with a zero coefficient:

$$2x^2 + 3x - 4 + 0x^2 + 0x - 4$$

To combine these polynomials, we need to ignore the terms with zero coefficients. The final answer is:

$$2x^2 + 3x - 8$$

Conclusion

Combining polynomials is an important skill in algebra. By following the steps outlined in this article, you can combine polynomials using addition and subtraction. Remember to identify like terms and combine them by adding or subtracting their coefficients. With practice, you will become proficient in combining polynomials.