Simplify The Expression:$\[ -8m^8 - 8m^6 + 20m^3 - 24m^2 \\]
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Introduction
Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the techniques involved to solve complex problems. In this article, we will focus on simplifying the given expression: . We will break down the expression into smaller parts, identify common factors, and use algebraic properties to simplify it.
Understanding the Expression
The given expression is a polynomial of degree 8, which means it has a variable (in this case, ) raised to the power of 8. The expression consists of four terms:
Identifying Common Factors
To simplify the expression, we need to identify common factors among the terms. A common factor is a term that divides each term in the expression without leaving a remainder. In this case, we can see that each term has a common factor of .
-8m^8 = -8(m^8)
-8m^6 = -8(m^6)
20m^3 = 20(m^3)
-24m^2 = -24(m^2)
Factoring Out the Common Factor
Now that we have identified the common factor, we can factor it out of each term. This will help us simplify the expression and make it easier to work with.
-8m^8 - 8m^6 + 20m^3 - 24m^2 = -8(m^8 + m^6) + 20(m^3) - 24(m^2)
Simplifying the Expression
Now that we have factored out the common factor, we can simplify the expression further. We can combine like terms and use algebraic properties to simplify the expression.
-8(m^8 + m^6) + 20(m^3) - 24(m^2) = -8m^8 - 8m^6 + 20m^3 - 24m^2
Using Algebraic Properties
We can use algebraic properties to simplify the expression further. One of the properties we can use is the distributive property, which states that:
We can use this property to simplify the expression:
-8(m^8 + m^6) = -8m^8 - 8m^6
Combining Like Terms
Now that we have simplified the expression using algebraic properties, we can combine like terms. Like terms are terms that have the same variable raised to the same power.
-8m^8 - 8m^6 + 20m^3 - 24m^2 = -8m^8 - 8m^6 + 20m^3 - 24m^2
Final Simplification
After combining like terms, we can simplify the expression further. We can use algebraic properties to simplify the expression:
-8m^8 - 8m^6 + 20m^3 - 24m^2 = -8(m^8 + m^6) + 20(m^3) - 24(m^2)
Conclusion
In this article, we have simplified the given expression using algebraic properties and techniques. We have identified common factors, factored them out, and combined like terms to simplify the expression. The final simplified expression is:
This expression is a simplified version of the original expression, and it can be used to solve complex problems in mathematics.
Frequently Asked Questions
Q: What is the degree of the simplified expression?
A: The degree of the simplified expression is 8.
Q: What is the leading coefficient of the simplified expression?
A: The leading coefficient of the simplified expression is -8.
Q: Can the simplified expression be factored further?
A: No, the simplified expression cannot be factored further.
References
Further Reading
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Introduction
In our previous article, we simplified the expression using algebraic properties and techniques. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.
Q&A
Q: What is the difference between simplifying and factoring an algebraic expression?
A: Simplifying an algebraic expression involves combining like terms and using algebraic properties to reduce the expression to its simplest form. Factoring an algebraic expression involves expressing it as a product of simpler expressions.
Q: How do I identify common factors in an algebraic expression?
A: To identify common factors, look for terms that have the same variable raised to the same power. You can also use the distributive property to factor out common factors.
Q: Can I simplify an algebraic expression by combining like terms?
A: Yes, you can simplify an algebraic expression by combining like terms. This involves adding or subtracting terms that have the same variable raised to the same power.
Q: What is the difference between a polynomial expression and an algebraic expression?
A: A polynomial expression is a type of algebraic expression that consists of terms with variables raised to non-negative integer powers. An algebraic expression, on the other hand, can consist of terms with variables raised to any power.
Q: How do I use algebraic properties to simplify an expression?
A: You can use algebraic properties such as the distributive property, the commutative property, and the associative property to simplify an expression. You can also use the properties of exponents to simplify expressions involving variables raised to powers.
Q: Can I simplify an expression by canceling out common factors?
A: Yes, you can simplify an expression by canceling out common factors. This involves dividing both sides of the equation by the common factor.
Q: What is the degree of an algebraic expression?
A: The degree of an algebraic expression is the highest power of the variable in the expression.
Q: How do I determine the leading coefficient of an algebraic expression?
A: The leading coefficient of an algebraic expression is the coefficient of the term with the highest power of the variable.
Q: Can I simplify an expression by using algebraic identities?
A: Yes, you can simplify an expression by using algebraic identities such as the Pythagorean identity or the difference of squares identity.
Examples
Example 1: Simplifying an Expression
Simplify the expression .
Solution:
- Combine like terms:
- Use algebraic properties:
Example 2: Factoring an Expression
Factor the expression .
Solution:
- Use the distributive property:
Conclusion
In this article, we have answered some frequently asked questions related to simplifying algebraic expressions. We have also provided examples of how to simplify and factor expressions using algebraic properties and techniques.
Frequently Asked Questions
Q: What is the difference between simplifying and factoring an algebraic expression?
A: Simplifying an algebraic expression involves combining like terms and using algebraic properties to reduce the expression to its simplest form. Factoring an algebraic expression involves expressing it as a product of simpler expressions.
Q: How do I identify common factors in an algebraic expression?
A: To identify common factors, look for terms that have the same variable raised to the same power. You can also use the distributive property to factor out common factors.
Q: Can I simplify an algebraic expression by combining like terms?
A: Yes, you can simplify an algebraic expression by combining like terms. This involves adding or subtracting terms that have the same variable raised to the same power.