Expand And Combine Like Terms.$\[ \left(2a^6 - 6a^3\right)^2 = \square \\]

Understanding the Concept of Expanding and Combining Like Terms

What are Like Terms?

In mathematics, like terms are terms that have the same variable raised to the same power. For example, 2x and 3x are like terms because they both have the variable x raised to the power of 1. Similarly, 4x^2 and 6x^2 are like terms because they both have the variable x raised to the power of 2.

What is Expanding and Combining Like Terms?

Expanding and combining like terms is a process of simplifying algebraic expressions by combining like terms. This involves multiplying out the brackets in an expression and then combining any like terms that are present.

Why is Expanding and Combining Like Terms Important?

Expanding and combining like terms is an important concept in algebra because it allows us to simplify complex expressions and solve equations more easily. It is a fundamental skill that is used in a wide range of mathematical applications, from solving quadratic equations to working with polynomial functions.

Expanding the Expression

To expand the expression $\left(2a^6 - 6a^3\right)^2$, we need to multiply out the brackets. This involves multiplying each term in the first bracket by each term in the second bracket.

Using the FOIL Method

The FOIL method is a technique for multiplying out brackets that involves multiplying the First terms, then the Outer terms, then the Inner terms, and finally the Last terms.

Applying the FOIL Method

Using the FOIL method, we can expand the expression as follows:

$\left(2a^6 - 6a^3\right)^2 = \left(2a^6\right)^2 - 2\left(2a^6\right)\left(6a^3\right) + \left(-6a^3\right)^2$

Simplifying the Expression

Now that we have expanded the expression, we can simplify it by combining like terms.

Combining Like Terms

To combine like terms, we need to identify the like terms in the expression and then add or subtract them accordingly.

Identifying Like Terms

In this case, the like terms are the terms that have the same variable raised to the same power. The like terms in this expression are $4a^{12}$, $-72a^9$, and $36a^6$.

Combining the Like Terms

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

Simplifying the Expression

After combining the like terms, the expression simplifies to:

$4a^{12} - 72a^9 + 36a^6$

Conclusion

In conclusion, expanding and combining like terms is an important concept in algebra that allows us to simplify complex expressions and solve equations more easily. By using the FOIL method and identifying like terms, we can expand and simplify expressions in a straightforward and efficient manner.

Example Problems

Problem 1

Expand and simplify the expression $\left(3x^2 - 2x + 1\right)^2$

Solution

Using the FOIL method, we can expand the expression as follows:

$\left(3x^2 - 2x + 1\right)^2 = \left(3x^2\right)^2 - 2\left(3x^2\right)\left(2x\right) + \left(-2x\right)^2 + 2\left(3x^2\right)\left(1\right) - 2\left(2x\right)\left(1\right) + \left(1\right)^2$

Simplifying the expression, we get:

$9x^4 - 12x^3 + 4x^2 + 6x^2 - 4x + 1$

Combining like terms, we get:

$9x^4 - 12x^3 + 10x^2 - 4x + 1$

Problem 2

Expand and simplify the expression $\left(2x^3 - 3x^2 + 1\right)^2$

Solution

Using the FOIL method, we can expand the expression as follows:

$\left(2x^3 - 3x^2 + 1\right)^2 = \left(2x^3\right)^2 - 2\left(2x^3\right)\left(3x^2\right) + \left(-3x^2\right)^2 + 2\left(2x^3\right)\left(1\right) - 2\left(3x^2\right)\left(1\right) + \left(1\right)^2$

Simplifying the expression, we get:

$4x^6 - 12x^5 + 9x^4 + 4x^3 - 6x^2 + 1$

Combining like terms, we get:

$4x^6 - 12x^5 + 9x^4 + 4x^3 - 6x^2 + 1$

Final Thoughts

In conclusion, expanding and combining like terms is an important concept in algebra that allows us to simplify complex expressions and solve equations more easily. By using the FOIL method and identifying like terms, we can expand and simplify expressions in a straightforward and efficient manner. With practice and patience, you can master this skill and become proficient in algebra.

Additional Resources

For further practice and review, you can try the following exercises:

• Expand and simplify the expression $\left(4x^2 - 3x + 2\right)^2$
• Expand and simplify the expression $\left(2x^3 - 4x^2 + 1\right)^2$
• Expand and simplify the expression $\left(3x^2 - 2x + 1\right)^3$

By practicing these exercises, you can improve your skills in expanding and combining like terms and become more confident in your ability to solve algebraic expressions.

Frequently Asked Questions

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 3x are like terms because they both have the variable x raised to the power of 1.

Q: What is expanding and combining like terms?

A: Expanding and combining like terms is a process of simplifying algebraic expressions by combining like terms. This involves multiplying out the brackets in an expression and then combining any like terms that are present.

Q: Why is expanding and combining like terms important?

A: Expanding and combining like terms is an important concept in algebra because it allows us to simplify complex expressions and solve equations more easily. It is a fundamental skill that is used in a wide range of mathematical applications, from solving quadratic equations to working with polynomial functions.

Q: How do I expand and combine like terms?

A: To expand and combine like terms, you need to follow these steps:

1. Multiply out the brackets in the expression.
2. Identify the like terms in the expression.
3. Combine the like terms by adding or subtracting them accordingly.

Q: What is the FOIL method?

A: The FOIL method is a technique for multiplying out brackets that involves multiplying the First terms, then the Outer terms, then the Inner terms, and finally the Last terms.

Q: How do I use the FOIL method?

A: To use the FOIL method, you need to follow these steps:

1. Multiply the First terms in the two brackets.
2. Multiply the Outer terms in the two brackets.
3. Multiply the Inner terms in the two brackets.
4. Multiply the Last terms in the two brackets.
5. Combine the like terms in the expression.

Q: What are some common mistakes to avoid when expanding and combining like terms?

A: Some common mistakes to avoid when expanding and combining like terms include:

• Forgetting to multiply out the brackets.
• Forgetting to identify the like terms in the expression.
• Forgetting to combine the like terms by adding or subtracting them accordingly.
• Making errors when multiplying out the brackets.

Q: How can I practice expanding and combining like terms?

A: You can practice expanding and combining like terms by working through exercises and problems that involve this concept. You can also try using online resources and tools to help you practice and review.

Advanced Questions

Q: How do I expand and combine like terms with negative coefficients?

A: To expand and combine like terms with negative coefficients, you need to follow the same steps as before, but you also need to take into account the negative sign. For example, if you have the expression (-2x)^2, you need to multiply out the brackets and then combine the like terms.

Q: How do I expand and combine like terms with fractional coefficients?

A: To expand and combine like terms with fractional coefficients, you need to follow the same steps as before, but you also need to take into account the fractional coefficient. For example, if you have the expression (1/2x)^2, you need to multiply out the brackets and then combine the like terms.

Q: How do I expand and combine like terms with variables in the denominator?

A: To expand and combine like terms with variables in the denominator, you need to follow the same steps as before, but you also need to take into account the variable in the denominator. For example, if you have the expression (x/2)^2, you need to multiply out the brackets and then combine the like terms.

Final Thoughts

Expanding and combining like terms is an important concept in algebra that allows us to simplify complex expressions and solve equations more easily. By following the steps outlined in this article and practicing regularly, you can become proficient in expanding and combining like terms and become more confident in your ability to solve algebraic expressions.

Additional Resources

For further practice and review, you can try the following exercises:

• Expand and simplify the expression (4x^2 - 3x + 2)^2
• Expand and simplify the expression (2x^3 - 4x^2 + 1)^2
• Expand and simplify the expression (3x^2 - 2x + 1)^3

By practicing these exercises, you can improve your skills in expanding and combining like terms and become more confident in your ability to solve algebraic expressions.