Simplify The Following Expression Completely:${ \frac{7x + 21}{x^2 - 7x - 30} }$Enter The Numerator And Denominator Separately In The Boxes Below. If The Denominator Is 1, Enter The Number 1. Do Not Leave Either Box Blank.Answer:Numerator:
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Introduction
Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the steps involved in simplifying complex expressions. In this article, we will simplify the given expression completely by factoring the numerator and denominator, and then canceling out any common factors.
The Given Expression
The given expression is:
Factor the Numerator
To simplify the expression, we need to factor the numerator. The numerator is a quadratic expression, and we can factor it by finding two numbers whose product is 21 and whose sum is 7.
7x + 21 = 7(x + 3)
Factor the Denominator
Next, we need to factor the denominator. The denominator is a quadratic expression, and we can factor it by finding two numbers whose product is -30 and whose sum is -7.
x^2 - 7x - 30 = (x - 10)(x + 3)
Simplify the Expression
Now that we have factored the numerator and denominator, we can simplify the expression by canceling out any common factors.
\frac{7(x + 3)}{(x - 10)(x + 3)} = \frac{7}{x - 10}
Conclusion
In this article, we simplified the given expression completely by factoring the numerator and denominator, and then canceling out any common factors. The final simplified expression is . This expression cannot be simplified further, and it is the completely simplified form of the given expression.
Discussion
Simplifying algebraic expressions is an essential skill in mathematics, and it's crucial to understand the steps involved in simplifying complex expressions. By factoring the numerator and denominator, and then canceling out any common factors, we can simplify expressions and make them easier to work with.
Tips and Tricks
- When simplifying expressions, always look for common factors in the numerator and denominator.
- Use factoring to simplify quadratic expressions.
- Cancel out any common factors to simplify the expression.
Related Topics
- Factoring quadratic expressions
- Simplifying algebraic expressions
- Canceling out common factors
Final Answer
The final answer is .
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Introduction
In our previous article, we simplified the given expression completely by factoring the numerator and denominator, and then canceling out any common factors. In this article, we will answer some frequently asked questions related to simplifying algebraic expressions.
Q&A
Q: What is the first step in simplifying an algebraic expression?
A: The first step in simplifying an algebraic expression is to look for any common factors in the numerator and denominator. If there are any common factors, we can cancel them out to simplify the expression.
Q: How do I factor a quadratic expression?
A: To factor a quadratic expression, we need to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term. We can then write the quadratic expression as a product of two binomials.
Q: What is the difference between simplifying an expression and canceling out common factors?
A: Simplifying an expression involves rewriting it in a simpler form, often by combining like terms or factoring out common factors. Canceling out common factors involves removing any common factors that appear in both the numerator and denominator.
Q: Can I simplify an expression if it has no common factors?
A: Yes, you can still simplify an expression even if it has no common factors. You can try to factor the numerator and denominator, or use other techniques such as combining like terms or using the distributive property.
Q: How do I know if an expression is completely simplified?
A: An expression is completely simplified if there are no more common factors that can be canceled out, and the numerator and denominator are in their simplest form.
Q: What are some common mistakes to avoid when simplifying expressions?
A: Some common mistakes to avoid when simplifying expressions include:
- Not canceling out common factors
- Not factoring the numerator and denominator
- Not combining like terms
- Not using the distributive property
Tips and Tricks
- Always look for common factors in the numerator and denominator.
- Use factoring to simplify quadratic expressions.
- Cancel out any common factors to simplify the expression.
- Combine like terms to simplify the expression.
- Use the distributive property to simplify the expression.
Related Topics
- Factoring quadratic expressions
- Simplifying algebraic expressions
- Canceling out common factors
- Combining like terms
- Using the distributive property
Final Answer
The final answer is .