Find The Value Of The Expression X 2 − 4 X − 8 \frac{x^2-4}{x-8} X − 8 X 2 − 4 When X = 4 X=4 X = 4 .

Jun 1, 2025 by ADMIN 104 views

Finding the Value of a Rational Expression

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Introduction

In algebra, rational expressions are a type of mathematical expression that involves a fraction with polynomials in both the numerator and denominator. These expressions can be simplified, evaluated, and manipulated using various techniques. In this article, we will focus on finding the value of a rational expression when a specific value is substituted for the variable.

Rational Expressions

A rational expression is a fraction that contains polynomials in both the numerator and denominator. It can be written in the form:

\frac{p(x)}{q(x)}

where $p(x)$ and $q(x)$ are polynomials. Rational expressions can be simplified, evaluated, and manipulated using various techniques, such as factoring, canceling, and substituting values.

Evaluating Rational Expressions

To evaluate a rational expression, we need to substitute the given value of the variable into the expression. This involves replacing the variable with the given value and simplifying the resulting expression.

Example

Let's consider the rational expression:

\frac{x^2-4}{x-8}

We are asked to find the value of this expression when $x=4$ . To do this, we need to substitute $x=4$ into the expression and simplify.

Substituting Values

To substitute a value into a rational expression, we need to replace the variable with the given value. In this case, we need to replace $x$ with $4$ .

\frac{(4)^2-4}{(4)-8}

Simplifying the Expression

Now that we have substituted the value, we need to simplify the resulting expression. This involves combining like terms and canceling out any common factors.

\frac{16-4}{-4}

\frac{12}{-4}

-3

Conclusion

In this article, we learned how to find the value of a rational expression when a specific value is substituted for the variable. We used the rational expression $\frac{x^2-4}{x-8}$ and substituted $x=4$ to find the value of the expression. We simplified the resulting expression to get the final answer.

Tips and Tricks

When substituting values into a rational expression, make sure to replace the variable with the given value.
When simplifying the expression, combine like terms and cancel out any common factors.
Use factoring and canceling techniques to simplify rational expressions.

Common Mistakes

Failing to substitute the value into the expression.
Not simplifying the expression correctly.
Not canceling out common factors.

Real-World Applications

Rational expressions have many real-world applications, such as:

Calculating the area and perimeter of shapes.
Finding the volume of solids.
Modeling population growth and decay.
Solving problems in physics and engineering.

Final Thoughts

In conclusion, finding the value of a rational expression when a specific value is substituted for the variable is an important skill in algebra. By following the steps outlined in this article, you can simplify rational expressions and find their values. Remember to substitute the value into the expression, simplify the resulting expression, and cancel out common factors. With practice and patience, you can become proficient in evaluating rational expressions.

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Introduction

In our previous article, we discussed how to find the value of a rational expression when a specific value is substituted for the variable. In this article, we will answer some frequently asked questions about rational expression evaluation.

Q&A

Q: What is a rational expression?

A: A rational expression is a fraction that contains polynomials in both the numerator and denominator. It can be written in the form:

\frac{p(x)}{q(x)}

where $p(x)$ and $q(x)$ are polynomials.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to combine like terms and cancel out any common factors. You can also use factoring and canceling techniques to simplify rational expressions.

Q: What is the difference between a rational expression and a rational number?

A: A rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. A rational expression is a fraction that contains polynomials in both the numerator and denominator.

Q: Can I simplify a rational expression by canceling out common factors?

A: Yes, you can simplify a rational expression by canceling out common factors. However, make sure that the common factors are not canceled out incorrectly.

Q: How do I evaluate a rational expression when the variable is a fraction?

A: To evaluate a rational expression when the variable is a fraction, you need to substitute the fraction into the expression and simplify.

Q: Can I use a calculator to evaluate a rational expression?

A: Yes, you can use a calculator to evaluate a rational expression. However, make sure that the calculator is set to the correct mode and that the expression is entered correctly.

Q: What is the order of operations for rational expressions?

A: The order of operations for rational expressions is the same as for numerical expressions:

Parentheses
Exponents
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

Q: Can I simplify a rational expression by combining like terms?

A: Yes, you can simplify a rational expression by combining like terms. However, make sure that the like terms are combined correctly.

Q: How do I know if a rational expression is undefined?

A: A rational expression is undefined if the denominator is equal to zero.

Q: Can I use a rational expression to model real-world problems?

A: Yes, you can use a rational expression to model real-world problems. Rational expressions can be used to model population growth and decay, chemical reactions, and other real-world phenomena.