Simplify The Following Expression Completely:$\[ \frac{x^2 + 12x + 20}{x^2 + 15x + 50} \\]Enter The Numerator And Denominator Separately In The Boxes Below. If The Denominator Is 1, Enter The Number 1. Do Not Leave Either Box
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Introduction
Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will focus on simplifying a given algebraic expression, which involves factoring the numerator and denominator, and then canceling out any common factors.
The Given Expression
The given expression is:
Our goal is to simplify this expression completely by factoring the numerator and denominator, and then canceling out any common factors.
Factoring the Numerator
To factor the numerator, we need to find two numbers whose product is 20 and whose sum is 12. These numbers are 5 and 4, since 5 × 4 = 20 and 5 + 4 = 9, but we need 12, so we will use 10 and 2, since 10 × 2 = 20 and 10 + 2 = 12.
However, we can factor the numerator as (x + 10)(x + 2).
Factoring the Denominator
To factor the denominator, we need to find two numbers whose product is 50 and whose sum is 15. These numbers are 10 and 5, since 10 × 5 = 50 and 10 + 5 = 15.
However, we can factor the denominator as (x + 10)(x + 5).
Canceling Out Common Factors
Now that we have factored the numerator and denominator, we can see that they have a common factor of (x + 10). We can cancel out this common factor by dividing both the numerator and denominator by (x + 10).
This gives us:
Simplifying the Expression
The expression is now simplified, and we can see that it is in the form of a rational expression.
Conclusion
In this article, we have simplified the given algebraic expression by factoring the numerator and denominator, and then canceling out any common factors. We have shown that the expression can be simplified to:
This is the final simplified form of the expression.
Final Answer
The final answer is:
Additional Tips and Tricks
- When simplifying algebraic expressions, it's essential to factor the numerator and denominator, and then cancel out any common factors.
- Make sure to check if the denominator is equal to zero before simplifying the expression.
- If the denominator is equal to zero, the expression is undefined, and you cannot simplify it further.
Frequently Asked Questions
- Q: What is the final simplified form of the expression? A: The final simplified form of the expression is .
- Q: How do I simplify an algebraic expression? A: To simplify an algebraic expression, you need to factor the numerator and denominator, and then cancel out any common factors.
- Q: What happens if the denominator is equal to zero? A: If the denominator is equal to zero, the expression is undefined, and you cannot simplify it further.
References
Related Articles
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Introduction
Simplifying algebraic expressions is a crucial skill in mathematics, and it can be a bit challenging, especially for beginners. In this article, we will address some of the most frequently asked questions related to simplifying algebraic expressions.
Q&A
Q: What is an algebraic expression?
A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.
Q: How do I simplify an algebraic expression?
A: To simplify an algebraic expression, you need to follow these steps:
- Factor the numerator and denominator.
- Cancel out any common factors.
- Simplify the resulting expression.
Q: What is factoring?
A: Factoring is the process of expressing an algebraic expression as a product of simpler expressions. For example, the expression x^2 + 5x + 6 can be factored as (x + 3)(x + 2).
Q: How do I factor an algebraic expression?
A: To factor an algebraic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the middle term. For example, to factor the expression x^2 + 5x + 6, you need to find two numbers whose product is 6 and whose sum is 5. These numbers are 2 and 3, since 2 × 3 = 6 and 2 + 3 = 5.
Q: What is a rational expression?
A: A rational expression is an algebraic expression that consists of a numerator and a denominator, and the denominator is not equal to zero.
Q: How do I simplify a rational expression?
A: To simplify a rational expression, you need to follow these steps:
- Factor the numerator and denominator.
- Cancel out any common factors.
- Simplify the resulting expression.
Q: What is a common factor?
A: A common factor is a factor that appears in both the numerator and the denominator of a rational expression.
Q: How do I cancel out common factors?
A: To cancel out common factors, you need to divide both the numerator and the denominator by the common factor.
Q: What happens if the denominator is equal to zero?
A: If the denominator is equal to zero, the expression is undefined, and you cannot simplify it further.
Q: How do I check if the denominator is equal to zero?
A: To check if the denominator is equal to zero, you need to set the denominator equal to zero and solve for the variable.
Q: What is the final simplified form of the expression?
A: The final simplified form of the expression is the expression that has been simplified as much as possible.
Additional Tips and Tricks
- Make sure to check if the denominator is equal to zero before simplifying the expression.
- If the denominator is equal to zero, the expression is undefined, and you cannot simplify it further.
- Use factoring to simplify algebraic expressions.
- Cancel out common factors to simplify rational expressions.
Frequently Asked Questions (FAQs)
- Q: What is an algebraic expression? A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.
- Q: How do I simplify an algebraic expression? A: To simplify an algebraic expression, you need to follow these steps: factor the numerator and denominator, cancel out any common factors, and simplify the resulting expression.
- Q: What is factoring? A: Factoring is the process of expressing an algebraic expression as a product of simpler expressions.