Simplify The Expression: X 2 + 10 X X − 1 \frac{x^2 + 10x}{x - 1} X − 1 X 2 + 10 X ​

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it is often used in various mathematical operations, such as solving equations and inequalities. In this article, we will focus on simplifying the given expression: $\frac{x^2 + 10x}{x - 1}$. We will use various techniques, such as factoring and canceling, to simplify the expression.

Understanding the Expression

The given expression is a rational expression, which is a fraction that contains variables and constants in the numerator and denominator. In this case, the numerator is $x^2 + 10x$, and the denominator is $x - 1$. To simplify the expression, we need to understand the properties of rational expressions.

Properties of Rational Expressions

Rational expressions have several properties that we need to understand:

• Cancellation: If a factor appears in both the numerator and denominator, we can cancel it out.
• Factoring: We can factor the numerator and denominator to simplify the expression.
• Simplifying: We can simplify the expression by canceling out common factors.

Factoring the Numerator

To simplify the expression, we can start by factoring the numerator. The numerator is $x^2 + 10x$, which can be factored as:

$x^2 + 10x = x(x + 10)$

Factoring the Denominator

The denominator is $x - 1$, which cannot be factored further.

Canceling Common Factors

Now that we have factored the numerator, we can cancel out common factors between the numerator and denominator. In this case, we can cancel out the factor $x$:

$\frac{x(x + 10)}{x - 1} = \frac{x + 10}{\frac{x - 1}{x}}$

However, we cannot cancel out the factor $x$ in the denominator because it would result in a division by zero.

Simplifying the Expression

To simplify the expression, we can use the fact that the denominator is $x - 1$. We can rewrite the expression as:

$\frac{x^2 + 10x}{x - 1} = \frac{x(x + 10)}{x - 1} = x + 10 + \frac{10x}{x - 1}$

However, this expression is still not simplified.

Using the Limit Definition of a Derivative

To simplify the expression, we can use the limit definition of a derivative. The limit definition of a derivative is:

$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

We can use this definition to simplify the expression:

$\frac{x^2 + 10x}{x - 1} = \lim_{h \to 0} \frac{(x + h)^2 + 10(x + h)}{(x + h) - 1}$

Expanding the numerator and denominator, we get:

$\lim_{h \to 0} \frac{x^2 + 2hx + h^2 + 10x + 10h}{hx + h - 1}$

Simplifying the expression, we get:

$\lim_{h \to 0} \frac{x^2 + 12x + h^2 + 10h}{hx + h - 1}$

Taking the limit as $h$ approaches zero, we get:

$\frac{x^2 + 12x}{x - 1}$

Final Simplification

To simplify the expression further, we can factor the numerator:

$\frac{x^2 + 12x}{x - 1} = \frac{x(x + 12)}{x - 1}$

However, we cannot cancel out the factor $x$ in the denominator because it would result in a division by zero.

Conclusion

In this article, we have simplified the expression $\frac{x^2 + 10x}{x - 1}$ using various techniques, such as factoring and canceling. We have also used the limit definition of a derivative to simplify the expression. The final simplified expression is $\frac{x(x + 12)}{x - 1}$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Simplifying Algebraic Expressions" by Math Open Reference

Further Reading

• [1] "Simplifying Rational Expressions" by Math Is Fun
• [2] "Factoring Algebraic Expressions" by Math Open Reference
• [3] "Limit Definition of a Derivative" by Khan Academy
  

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Introduction

In our previous article, we simplified the expression $\frac{x^2 + 10x}{x - 1}$ using various techniques, such as factoring and canceling. We also used the limit definition of a derivative to simplify the expression. In this article, we will answer some frequently asked questions (FAQs) related to simplifying the expression.

Q&A

Q: What is the final simplified expression of $\frac{x^2 + 10x}{x - 1}$?

A: The final simplified expression of $\frac{x^2 + 10x}{x - 1}$ is $\frac{x(x + 12)}{x - 1}$.

Q: Can I cancel out the factor $x$ in the denominator?

A: No, you cannot cancel out the factor $x$ in the denominator because it would result in a division by zero.

Q: What is the limit definition of a derivative?

A: The limit definition of a derivative is:

$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Q: How do I use the limit definition of a derivative to simplify the expression?

A: To use the limit definition of a derivative to simplify the expression, you can substitute the expression into the limit definition and simplify.

Q: What are some common techniques used to simplify algebraic expressions?

A: Some common techniques used to simplify algebraic expressions include:

• Factoring: This involves expressing the expression as a product of simpler expressions.
• Canceling: This involves canceling out common factors between the numerator and denominator.
• Simplifying: This involves simplifying the expression by combining like terms.

Q: How do I factor the numerator of the expression?

A: To factor the numerator of the expression, you can look for common factors and group them together.

Q: How do I cancel out common factors between the numerator and denominator?

A: To cancel out common factors between the numerator and denominator, you can look for common factors and cancel them out.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

• Canceling out factors that are not common to both the numerator and denominator.
• Failing to simplify the expression by combining like terms.
• Using the wrong techniques to simplify the expression.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to simplifying the expression $\frac{x^2 + 10x}{x - 1}$. We have also provided some common techniques used to simplify algebraic expressions and some common mistakes to avoid.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Simplifying Algebraic Expressions" by Math Open Reference

Further Reading

• [1] "Simplifying Rational Expressions" by Math Is Fun
• [2] "Factoring Algebraic Expressions" by Math Open Reference
• [3] "Limit Definition of a Derivative" by Khan Academy