Combine Like Terms To Simplify The Following Polynomial: $2x + 3z - X + 5z$A) $x + 8z$ B) $11xz$ C) $3x - 7z$ D) $2x^2 + 3z^2$

Introduction

In algebra, polynomials are expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. Simplifying polynomials is an essential skill in mathematics, as it helps to make complex expressions more manageable and easier to work with. In this article, we will focus on combining like terms to simplify a given polynomial.

What are Like Terms?

Like terms are terms in a polynomial that have the same variable(s) raised to the same power. For example, in the polynomial $2x + 3x$, the terms $2x$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 1.

Step 1: Identify Like Terms

To simplify a polynomial, we need to identify the like terms. In the given polynomial $2x + 3z - x + 5z$, we can see that the terms $2x$ and $-x$ are like terms because they both have the variable $x$ raised to the power of 1. Similarly, the terms $3z$ and $5z$ are like terms because they both have the variable $z$ raised to the power of 1.

Step 2: Combine Like Terms

Once we have identified the like terms, we can combine them by adding or subtracting their coefficients. In the given polynomial, we can combine the like terms $2x$ and $-x$ by adding their coefficients: $2x - x = x$. Similarly, we can combine the like terms $3z$ and $5z$ by adding their coefficients: $3z + 5z = 8z$.

Step 3: Write the Simplified Polynomial

After combining the like terms, we can write the simplified polynomial by adding the combined terms. In this case, the simplified polynomial is $x + 8z$.

Conclusion

In conclusion, combining like terms is an essential skill in algebra that helps to simplify polynomials. By identifying like terms and combining them, we can make complex expressions more manageable and easier to work with. In this article, we have seen how to simplify a polynomial by combining like terms.

Example Problems

Problem 1

Simplify the polynomial $3x + 2y - 2x + 4y$.

Solution

To simplify the polynomial, we need to identify the like terms. In this case, the like terms are $3x$ and $-2x$, and $2y$ and $4y$. We can combine these like terms by adding their coefficients:

$3x - 2x = x$ $2y + 4y = 6y$

The simplified polynomial is $x + 6y$.

Problem 2

Simplify the polynomial $2x^2 + 3x - 4x^2 + 5x$.

Solution

To simplify the polynomial, we need to identify the like terms. In this case, the like terms are $2x^2$ and $-4x^2$, and $3x$ and $5x$. We can combine these like terms by adding their coefficients:

$2x^2 - 4x^2 = -2x^2$ $3x + 5x = 8x$

The simplified polynomial is $-2x^2 + 8x$.

Tips and Tricks

• When combining like terms, make sure to add or subtract the coefficients correctly.
• When simplifying a polynomial, make sure to combine all the like terms.
• When writing the simplified polynomial, make sure to add the combined terms correctly.

Common Mistakes

• Not identifying like terms correctly.
• Not combining like terms correctly.
• Not writing the simplified polynomial correctly.

Conclusion

Q: What are like terms in a polynomial?

A: Like terms are terms in a polynomial that have the same variable(s) raised to the same power. For example, in the polynomial $2x + 3x$, the terms $2x$ and $3x$ are like terms because they both have the variable $x$ raised to the power of 1.

Q: How do I identify like terms in a polynomial?

A: To identify like terms, look for terms that have the same variable(s) raised to the same power. For example, in the polynomial $2x^2 + 3x - 4x^2 + 5x$, the terms $2x^2$ and $-4x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I combine like terms in a polynomial?

A: To combine like terms, add or subtract their coefficients. For example, in the polynomial $2x + 3x - 2x + 4x$, the like terms are $2x$, $3x$, $-2x$, and $4x$. We can combine these like terms by adding their coefficients:

$2x + 3x = 5x$ $-2x + 4x = 2x$

The simplified polynomial is $5x + 2x = 7x$.

Q: What is the difference between combining like terms and simplifying a polynomial?

A: Combining like terms is the process of adding or subtracting the coefficients of like terms in a polynomial. Simplifying a polynomial is the process of combining like terms and removing any unnecessary terms. For example, in the polynomial $2x + 3x - 2x + 4x$, we can combine the like terms to get $7x$. This is an example of combining like terms. However, if we simplify the polynomial further by removing the unnecessary terms, we get $7x$.

Q: Can I simplify a polynomial with variables raised to different powers?

A: Yes, you can simplify a polynomial with variables raised to different powers. For example, in the polynomial $2x^2 + 3x - 4x^2 + 5x$, we can combine the like terms $2x^2$ and $-4x^2$ to get $-2x^2$. We can also combine the like terms $3x$ and $5x$ to get $8x$. The simplified polynomial is $-2x^2 + 8x$.

Q: Can I simplify a polynomial with variables raised to the same power but with different coefficients?

A: Yes, you can simplify a polynomial with variables raised to the same power but with different coefficients. For example, in the polynomial $2x^2 + 3x^2 - 4x^2 + 5x^2$, we can combine the like terms $2x^2$, $3x^2$, $-4x^2$, and $5x^2$ to get $6x^2$. The simplified polynomial is $6x^2$.

Q: Can I simplify a polynomial with variables raised to different powers and with different coefficients?

A: Yes, you can simplify a polynomial with variables raised to different powers and with different coefficients. For example, in the polynomial $2x^2 + 3x - 4x^2 + 5x$, we can combine the like terms $2x^2$ and $-4x^2$ to get $-2x^2$. We can also combine the like terms $3x$ and $5x$ to get $8x$. The simplified polynomial is $-2x^2 + 8x$.

Q: Can I simplify a polynomial with variables raised to the same power but with different coefficients and with variables raised to different powers?

A: Yes, you can simplify a polynomial with variables raised to the same power but with different coefficients and with variables raised to different powers. For example, in the polynomial $2x^2 + 3x^2 - 4x^2 + 5x^2 + 2x - 3x$, we can combine the like terms $2x^2$, $3x^2$, $-4x^2$, and $5x^2$ to get $6x^2$. We can also combine the like terms $2x$ and $-3x$ to get $-x$. The simplified polynomial is $6x^2 - x$.

Conclusion

In conclusion, simplifying polynomials is an essential skill in algebra that helps to make complex expressions more manageable and easier to work with. By combining like terms and removing any unnecessary terms, we can simplify polynomials and make them easier to understand. In this article, we have seen some frequently asked questions and answers about simplifying polynomials. We have also seen some examples of how to simplify polynomials with variables raised to different powers and with different coefficients.