Combine Any Like Terms In The Expression. If There Are No Like Terms, Rewrite The Expression.$31 F^2 + 44 F^3 - 6 F^2$

Understanding Like Terms

In algebra, like terms are expressions that have the same variable raised to the same power. For example, in the expression $3x^2 + 2x^2$, the terms $3x^2$ and $2x^2$ are like terms because they both have the variable $x$ raised to the power of 2. On the other hand, the terms $3x^2$ and $2x$ are not like terms because they have different powers of the variable.

Combining Like Terms

When we have like terms in an expression, we can combine them by adding or subtracting their coefficients. The coefficient of a term is the number that is multiplied by the variable. For example, in the term $3x^2$, the coefficient is 3.

To combine like terms, we follow these steps:

1. Identify the like terms in the expression.
2. Add or subtract the coefficients of the like terms.
3. Write the resulting term with the combined coefficient and the same variable raised to the same power.

Example: Combining Like Terms in the Expression $31 f^2 + 44 f^3 - 6 f^2$

Let's apply the steps above to the expression $31 f^2 + 44 f^3 - 6 f^2$. The first step is to identify the like terms in the expression.

In this expression, the like terms are $31 f^2$ and $-6 f^2$ because they both have the variable $f$ raised to the power of 2. The term $44 f^3$ is not a like term because it has a different power of the variable.

The next step is to add or subtract the coefficients of the like terms. In this case, we add the coefficients of the like terms:

$31 f^2 + (-6 f^2) = (31 - 6) f^2 = 25 f^2$

So, the expression $31 f^2 + 44 f^3 - 6 f^2$ can be simplified to $25 f^2 + 44 f^3$.

Simplifying the Expression

Now that we have combined the like terms, we can simplify the expression further by rewriting it in a more compact form.

The expression $25 f^2 + 44 f^3$ can be rewritten as $44 f^3 + 25 f^2$. This is because the order of the terms does not affect the value of the expression.

Conclusion

In this article, we learned how to combine like terms in an algebraic expression. We identified the like terms in the expression $31 f^2 + 44 f^3 - 6 f^2$, added or subtracted their coefficients, and wrote the resulting term with the combined coefficient and the same variable raised to the same power. We also simplified the expression by rewriting it in a more compact form.

Tips and Tricks

• When combining like terms, make sure to add or subtract the coefficients of the like terms.
• The order of the terms does not affect the value of the expression.
• When simplifying an expression, make sure to rewrite it in a more compact form.

Common Mistakes to Avoid

• Not identifying like terms in the expression.
• Not adding or subtracting the coefficients of the like terms.
• Not rewriting the expression in a more compact form.

Real-World Applications

Combining like terms is an essential skill in algebra that has many real-world applications. For example, in physics, combining like terms is used to simplify complex equations that describe the motion of objects. In engineering, combining like terms is used to simplify complex equations that describe the behavior of electrical circuits.

Practice Problems

1. Combine the like terms in the expression $2x^2 + 5x^2 - 3x^2$.
2. Combine the like terms in the expression $4y^3 + 2y^3 - 6y^3$.
3. Combine the like terms in the expression $3z^2 + 2z^2 - 5z^2$.

Answer Key

1. $(2 + 5 - 3) x^2 = 4 x^2$
2. $(4 + 2 - 6) y^3 = 0$
3. $(3 + 2 - 5) z^2 = 0$

Conclusion

$$

Frequently Asked Questions

Q: What are like terms in algebra?

A: Like terms are expressions that have the same variable raised to the same power. For example, in the expression $3x^2 + 2x^2$, the terms $3x^2$ and $2x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I identify like terms in an expression?

A: To identify like terms, look for expressions that have the same variable raised to the same power. For example, in the expression $2x^2 + 5x^2 - 3x^2$, the terms $2x^2$, $5x^2$, and $-3x^2$ are like terms because they all have the variable $x$ raised to the power of 2.

Q: How do I combine like terms in an expression?

A: To combine like terms, add or subtract the coefficients of the like terms. For example, in the expression $2x^2 + 5x^2 - 3x^2$, the coefficients are 2, 5, and -3. Adding these coefficients gives us $2 + 5 - 3 = 4$. Therefore, the expression $2x^2 + 5x^2 - 3x^2$ can be simplified to $4x^2$.

Q: What if I have a term with a variable raised to a power and a term with a variable raised to a different power?

A: If you have a term with a variable raised to a power and a term with a variable raised to a different power, they are not like terms. For example, in the expression $2x^2 + 3x$, the terms $2x^2$ and $3x$ are not like terms because they have different powers of the variable.

Q: Can I combine like terms in an expression with multiple variables?

A: Yes, you can combine like terms in an expression with multiple variables. For example, in the expression $2xy^2 + 3xy^2 - 4xy^2$, the terms $2xy^2$, $3xy^2$, and $-4xy^2$ are like terms because they all have the variables $x$ and $y$ raised to the same powers.

Q: How do I simplify an expression with like terms?

A: To simplify an expression with like terms, combine the like terms by adding or subtracting their coefficients. Then, rewrite the expression in a more compact form. For example, in the expression $2x^2 + 5x^2 - 3x^2$, combining the like terms gives us $4x^2$. Therefore, the expression $2x^2 + 5x^2 - 3x^2$ can be simplified to $4x^2$.

Q: What if I have an expression with no like terms?

A: If you have an expression with no like terms, the expression cannot be simplified further. For example, in the expression $2x^2 + 3x - 4$, there are no like terms, so the expression cannot be simplified further.

Q: Can I use a calculator to combine like terms?

A: Yes, you can use a calculator to combine like terms. However, it's always a good idea to check your work by hand to make sure you get the correct answer.

Q: How do I know if I have combined like terms correctly?

A: To check if you have combined like terms correctly, make sure to add or subtract the coefficients of the like terms correctly. You can also use a calculator to check your work.

Q: What if I make a mistake when combining like terms?

A: If you make a mistake when combining like terms, don't worry! Just go back and recheck your work. Make sure to add or subtract the coefficients of the like terms correctly.

Q: Can I combine like terms in an expression with negative coefficients?

A: Yes, you can combine like terms in an expression with negative coefficients. For example, in the expression $-2x^2 + 3x^2 - 4x^2$, the coefficients are -2, 3, and -4. Adding these coefficients gives us $-2 + 3 - 4 = -3$. Therefore, the expression $-2x^2 + 3x^2 - 4x^2$ can be simplified to $-3x^2$.

Q: Can I combine like terms in an expression with fractions as coefficients?

A: Yes, you can combine like terms in an expression with fractions as coefficients. For example, in the expression $\frac{1}{2}x^2 + \frac{3}{2}x^2 - \frac{4}{2}x^2$, the coefficients are $\frac{1}{2}$, $\frac{3}{2}$, and $-\frac{4}{2}$. Adding these coefficients gives us $\frac{1}{2} + \frac{3}{2} - \frac{4}{2} = -\frac{1}{2}$. Therefore, the expression $\frac{1}{2}x^2 + \frac{3}{2}x^2 - \frac{4}{2}x^2$ can be simplified to $-\frac{1}{2}x^2$.

Q: Can I combine like terms in an expression with variables as coefficients?

A: No, you cannot combine like terms in an expression with variables as coefficients. For example, in the expression $2x + 3x$, the coefficients are 2 and 3, which are variables. You cannot add or subtract variables, so the expression $2x + 3x$ cannot be simplified further.

Q: Can I combine like terms in an expression with exponents as coefficients?

A: No, you cannot combine like terms in an expression with exponents as coefficients. For example, in the expression $2x^2 + 3x^2$, the coefficients are 2 and 3, which are exponents. You cannot add or subtract exponents, so the expression $2x^2 + 3x^2$ cannot be simplified further.

Q: Can I combine like terms in an expression with radicals as coefficients?

A: No, you cannot combine like terms in an expression with radicals as coefficients. For example, in the expression $2\sqrt{x} + 3\sqrt{x}$, the coefficients are 2 and 3, which are radicals. You cannot add or subtract radicals, so the expression $2\sqrt{x} + 3\sqrt{x}$ cannot be simplified further.

Q: Can I combine like terms in an expression with absolute values as coefficients?

A: No, you cannot combine like terms in an expression with absolute values as coefficients. For example, in the expression $2|x| + 3|x|$, the coefficients are 2 and 3, which are absolute values. You cannot add or subtract absolute values, so the expression $2|x| + 3|x|$ cannot be simplified further.

Q: Can I combine like terms in an expression with complex numbers as coefficients?

A: No, you cannot combine like terms in an expression with complex numbers as coefficients. For example, in the expression $2(3 + 4i) + 3(3 + 4i)$, the coefficients are 2 and 3, which are complex numbers. You cannot add or subtract complex numbers, so the expression $2(3 + 4i) + 3(3 + 4i)$ cannot be simplified further.

Q: Can I combine like terms in an expression with matrices as coefficients?

A: No, you cannot combine like terms in an expression with matrices as coefficients. For example, in the expression $2\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + 3\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, the coefficients are 2 and 3, which are matrices. You cannot add or subtract matrices, so the expression $2\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + 3\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ cannot be simplified further.

Q: Can I combine like terms in an expression with vectors as coefficients?

A: No, you cannot combine like terms in an expression with vectors as coefficients. For example, in the expression $2\begin{bmatrix} 1 \\ 2 \end{bmatrix} + 3\begin{bmatrix} 1 \\ 2 \end{bmatrix}$, the coefficients are 2 and 3, which are vectors. You cannot add or subtract vectors, so the expression $2\begin{bmatrix} 1 \\ 2 \end{bmatrix} + 3\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ cannot be simplified further.

Q: Can I combine like terms in an expression with functions as coefficients?

A: No, you cannot combine like terms in an expression with functions as coefficients. For example, in the expression $2f(x) + 3f(x)$, the coefficients are 2 and 3, which are functions. You cannot add or subtract functions, so the expression $2f(x) + 3f(x)$ cannot be simplified further