Simplify:\[$(x-3)(x^2-4x-7)\$\]A. \[$x^3-3x^2+12x+21\$\] B. \[$x^3-7x^2+19x+21\$\] C. \[$x^3-7x^2+5x+21\$\] D. \[$x^3+x^2-5x-21\$\]
Introduction
In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. One of the most common techniques used to simplify expressions is the distributive property, which allows us to multiply each term in one expression by each term in another expression. In this article, we will use the distributive property to simplify the given expression {(x-3)(x^2-4x-7)$}$.
Understanding the Distributive Property
The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c, the following equation holds:
a(b + c) = ab + ac
This property can be extended to more than two terms, and it is a powerful tool for simplifying expressions. In the context of the given expression, we will use the distributive property to multiply each term in the first expression (x - 3) by each term in the second expression (x^2 - 4x - 7).
Applying the Distributive Property
To simplify the given expression, we will apply the distributive property as follows:
{(x-3)(x^2-4x-7)$ = x(x^2-4x-7) - 3(x^2-4x-7)}$
Expanding the Terms
Now, we will expand each term in the expression using the distributive property:
{x(x^2-4x-7) - 3(x^2-4x-7)$ = x^3 - 4x^2 - 7x - 3x^2 + 12x + 21}$
Combining Like Terms
The next step is to combine like terms in the expression. Like terms are terms that have the same variable and exponent. In this case, we have the following like terms:
- x^3
- -4x^2
- -3x^2
- -7x
- 12x
- 21
Combining these like terms, we get:
{x^3 - 7x^2 + 5x + 21}$
Conclusion
In conclusion, we have successfully simplified the given expression {(x-3)(x^2-4x-7)$ using the distributive property. The simplified expression is [x^3 - 7x^2 + 5x + 21}. This expression is the correct answer to the given problem.
Final Answer
The final answer to the problem is:
{x^3 - 7x^2 + 5x + 21}$
This answer is option C in the given multiple-choice question.
Discussion
The given problem is a classic example of how to simplify expressions using the distributive property. The distributive property is a fundamental concept in algebra that helps us simplify expressions and solve equations. In this article, we have demonstrated how to apply the distributive property to simplify the given expression.
Tips and Tricks
Here are some tips and tricks to help you simplify expressions using the distributive property:
- Make sure to apply the distributive property to each term in the expression.
- Combine like terms carefully to avoid making mistakes.
- Use the distributive property to simplify expressions with multiple terms.
- Practice simplifying expressions using the distributive property to become more comfortable with the concept.
Common Mistakes
Here are some common mistakes to avoid when simplifying expressions using the distributive property:
- Failing to apply the distributive property to each term in the expression.
- Not combining like terms carefully.
- Making mistakes when multiplying terms.
- Not checking the final answer for errors.
Real-World Applications
The distributive property has many real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, the distributive property is used to simplify complex mathematical expressions that describe the behavior of electrical circuits. In physics, the distributive property is used to simplify expressions that describe the motion of objects. In computer science, the distributive property is used to simplify expressions that describe the behavior of algorithms.
Conclusion
In conclusion, the distributive property is a powerful tool for simplifying expressions in algebra. By applying the distributive property, we can simplify complex expressions and solve equations. In this article, we have demonstrated how to apply the distributive property to simplify the given expression. We have also provided tips and tricks to help you simplify expressions using the distributive property, as well as common mistakes to avoid. Finally, we have discussed the real-world applications of the distributive property in fields such as engineering, physics, and computer science.
Introduction
In our previous article, we simplified the given expression [$(x-3)(x^2-4x-7)$ using the distributive property. In this article, we will answer some common questions related to simplifying expressions using the distributive property.
Q: What is the distributive property?
A: The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c, the following equation holds:
a(b + c) = ab + ac
This property can be extended to more than two terms, and it is a powerful tool for simplifying expressions.
Q: How do I apply the distributive property to simplify expressions?
A: To apply the distributive property, you need to multiply each term in one expression by each term in another expression. For example, to simplify the expression [$(x-3)(x^2-4x-7)$, you would multiply each term in the first expression (x - 3) by each term in the second expression (x^2 - 4x - 7).
Q: What are like terms?
A: Like terms are terms that have the same variable and exponent. For example, in the expression [$x^3 - 7x^2 + 5x + 21$, the terms -7x^2 and 5x are like terms because they have the same variable (x) and exponent (2).
Q: How do I combine like terms?
A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, in the expression [$x^3 - 7x^2 + 5x + 21$, the like terms -7x^2 and 5x can be combined by adding their coefficients: -7x^2 + 5x = -2x^2.
Q: What are some common mistakes to avoid when simplifying expressions using the distributive property?
A: Some common mistakes to avoid when simplifying expressions using the distributive property include:
- Failing to apply the distributive property to each term in the expression.
- Not combining like terms carefully.
- Making mistakes when multiplying terms.
- Not checking the final answer for errors.
Q: How do I check my answer for errors?
A: To check your answer for errors, you need to plug in some values for the variable (x) and see if the expression simplifies to the correct answer. For example, if you simplify the expression [x^3 - 7x^2 + 5x + 21$, you can plug in x = 1 and see if the expression simplifies to the correct answer: (1 - 3)(1^2 - 4(1) - 7) = (-2)(-10) = 20, which is not equal to the correct answer. This indicates that there is an error in the simplification.
Q: What are some real-world applications of the distributive property?
A: The distributive property has many real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, the distributive property is used to simplify complex mathematical expressions that describe the behavior of electrical circuits. In physics, the distributive property is used to simplify expressions that describe the motion of objects. In computer science, the distributive property is used to simplify expressions that describe the behavior of algorithms.
Q: How can I practice simplifying expressions using the distributive property?
A: You can practice simplifying expressions using the distributive property by working through some examples and exercises. You can also try simplifying expressions on your own and then checking your answers for errors.
Q: What are some tips for simplifying expressions using the distributive property?
A: Some tips for simplifying expressions using the distributive property include:
- Make sure to apply the distributive property to each term in the expression.
- Combine like terms carefully to avoid making mistakes.
- Use the distributive property to simplify expressions with multiple terms.
- Practice simplifying expressions using the distributive property to become more comfortable with the concept.
Conclusion
In conclusion, the distributive property is a powerful tool for simplifying expressions in algebra. By applying the distributive property, we can simplify complex expressions and solve equations. In this article, we have answered some common questions related to simplifying expressions using the distributive property, including how to apply the distributive property, what are like terms, how to combine like terms, and how to check your answer for errors. We have also discussed some real-world applications of the distributive property and provided some tips for simplifying expressions using the distributive property.