Answer Two Questions About The Following Rational Expression. { \frac{x 2-4x+4}{x 2+5x-14}$}$1. What Is An Equivalent Expression In Lowest Terms? { \square$}$2. What Values Of { X$}$ Must We Exclude From The Domains Of

Introduction

Rational expressions are a fundamental concept in algebra, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying rational expressions, focusing on the given expression: $\frac{x^2-4x+4}{x^2+5x-14}$. We will answer two questions related to this expression: (1) what is an equivalent expression in lowest terms, and (2) what values of $x$ must we exclude from the domains of the original and simplified expressions.

Question 1: Equivalent Expression in Lowest Terms

To simplify the given rational expression, we need to factor both the numerator and the denominator. The numerator can be factored as a perfect square trinomial:

$$x^2-4x+4 = (x-2)^2$$

The denominator can be factored as a quadratic expression:

$$x^2+5x-14 = (x+7)(x-2)$$

Now, we can rewrite the original expression using these factored forms:

$$\frac{(x-2)^2}{(x+7)(x-2)}$$

We can see that the term $(x-2)$ appears in both the numerator and the denominator. We can cancel out this common factor, but we need to be careful not to cancel out any factors that would make the denominator zero. In this case, we can cancel out the $(x-2)$ term:

$$\frac{(x-2)^2}{(x+7)(x-2)} = \frac{x-2}{x+7}$$

Therefore, an equivalent expression in lowest terms is $\frac{x-2}{x+7}$.

Question 2: Excluded Values of $x$

When simplifying rational expressions, we need to be aware of the values of $x$ that would make the denominator zero. These values are called excluded values, and they must be excluded from the domain of the expression.

In the original expression, the denominator is $(x+7)(x-2)$. We can see that the denominator would be zero when $x=-7$ or $x=2$. Therefore, these values of $x$ must be excluded from the domain of the original expression.

Similarly, when we simplified the expression to $\frac{x-2}{x+7}$, we need to exclude the same values of $x$ from the domain of the simplified expression.

Conclusion

In conclusion, we have simplified the given rational expression $\frac{x^2-4x+4}{x^2+5x-14}$ to its equivalent expression in lowest terms: $\frac{x-2}{x+7}$. We have also identified the values of $x$ that must be excluded from the domains of both the original and simplified expressions: $x=-7$ and $x=2$.

Final Answer

1. An equivalent expression in lowest terms is $\frac{x-2}{x+7}$.
2. The values of $x$ that must be excluded from the domains of the original and simplified expressions are $x=-7$ and $x=2$.

Additional Tips and Tricks

• When simplifying rational expressions, always factor the numerator and denominator.
• Be careful not to cancel out any factors that would make the denominator zero.
• Identify the excluded values of $x$ and exclude them from the domain of the expression.
• Practice simplifying rational expressions to become more comfortable with the process.

Common Mistakes to Avoid

• Canceling out factors that would make the denominator zero.
• Failing to identify excluded values of $x$.
• Not factoring the numerator and denominator.
• Not simplifying the expression to its lowest terms.

Real-World Applications

Rational expressions are used in a variety of real-world applications, including:

• Physics: to describe the motion of objects.
• Engineering: to design and analyze systems.
• Economics: to model economic systems.
• Computer Science: to develop algorithms and data structures.

Q: What is a rational expression?

A: A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to factor the numerator and denominator, and then cancel out any common factors.

Q: What is factoring?

A: Factoring is the process of expressing a polynomial as a product of simpler polynomials.

Q: How do I factor a polynomial?

A: There are several methods for factoring polynomials, including:

• Factoring out the greatest common factor (GCF)
• Factoring by grouping
• Factoring quadratics
• Factoring polynomials with rational roots

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides each term of a polynomial.

Q: How do I find the GCF of a polynomial?

A: To find the GCF of a polynomial, you need to list the factors of each term and find the greatest common factor.

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, whereas a rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator.

Q: Can I simplify a rational expression with a variable in the denominator?

A: Yes, you can simplify a rational expression with a variable in the denominator, but you need to be careful not to cancel out any factors that would make the denominator zero.

Q: How do I know if a rational expression is in its simplest form?

A: A rational expression is in its simplest form if it has no common factors in the numerator and denominator.

Q: Can I simplify a rational expression with a negative exponent?

A: Yes, you can simplify a rational expression with a negative exponent, but you need to follow the rules of exponents.

Q: How do I simplify a rational expression with a fraction in the denominator?

A: To simplify a rational expression with a fraction in the denominator, you need to multiply the numerator and denominator by the reciprocal of the fraction in the denominator.

Q: Can I simplify a rational expression with a variable in the numerator and denominator?

A: Yes, you can simplify a rational expression with a variable in the numerator and denominator, but you need to be careful not to cancel out any factors that would make the denominator zero.

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

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

Q: Can I simplify a rational expression with a complex number in the denominator?

A: Yes, you can simplify a rational expression with a complex number in the denominator, but you need to follow the rules of complex numbers.

Q: How do I simplify a rational expression with a radical in the denominator?

A: To simplify a rational expression with a radical in the denominator, you need to multiply the numerator and denominator by the conjugate of the radical in the denominator.

Q: Can I simplify a rational expression with a trigonometric function in the denominator?

A: Yes, you can simplify a rational expression with a trigonometric function in the denominator, but you need to follow the rules of trigonometric functions.

Q: How do I simplify a rational expression with a logarithmic function in the denominator?

A: To simplify a rational expression with a logarithmic function in the denominator, you need to multiply the numerator and denominator by the reciprocal of the logarithmic function in the denominator.

Q: Can I simplify a rational expression with a polynomial in the denominator?

A: Yes, you can simplify a rational expression with a polynomial in the denominator, but you need to follow the rules of polynomials.

Q: How do I simplify a rational expression with a rational function in the denominator?

A: To simplify a rational expression with a rational function in the denominator, you need to multiply the numerator and denominator by the reciprocal of the rational function in the denominator.

Q: Can I simplify a rational expression with a transcendental function in the denominator?

A: Yes, you can simplify a rational expression with a transcendental function in the denominator, but you need to follow the rules of transcendental functions.

Q: How do I simplify a rational expression with a mixed fraction in the denominator?

A: To simplify a rational expression with a mixed fraction in the denominator, you need to convert the mixed fraction to an improper fraction and then simplify.

Q: Can I simplify a rational expression with a decimal in the denominator?

A: Yes, you can simplify a rational expression with a decimal in the denominator, but you need to follow the rules of decimals.

Q: How do I simplify a rational expression with a percentage in the denominator?

A: To simplify a rational expression with a percentage in the denominator, you need to convert the percentage to a decimal and then simplify.

Q: Can I simplify a rational expression with a fraction in the numerator and denominator?

A: Yes, you can simplify a rational expression with a fraction in the numerator and denominator, but you need to follow the rules of fractions.

Q: How do I simplify a rational expression with a variable in the numerator and denominator?

A: To simplify a rational expression with a variable in the numerator and denominator, you need to factor the numerator and denominator and then cancel out any common factors.

Q: Can I simplify a rational expression with a complex fraction in the denominator?

A: Yes, you can simplify a rational expression with a complex fraction in the denominator, but you need to follow the rules of complex fractions.

Q: How do I simplify a rational expression with a nested fraction in the denominator?

A: To simplify a rational expression with a nested fraction in the denominator, you need to multiply the numerator and denominator by the reciprocal of the nested fraction in the denominator.

Q: Can I simplify a rational expression with a fraction in the numerator and a variable in the denominator?

A: Yes, you can simplify a rational expression with a fraction in the numerator and a variable in the denominator, but you need to follow the rules of fractions and variables.

Q: How do I simplify a rational expression with a variable in the numerator and a fraction in the denominator?

A: To simplify a rational expression with a variable in the numerator and a fraction in the denominator, you need to factor the numerator and denominator and then cancel out any common factors.

Q: Can I simplify a rational expression with a complex number in the numerator and denominator?

A: Yes, you can simplify a rational expression with a complex number in the numerator and denominator, but you need to follow the rules of complex numbers.

Q: How do I simplify a rational expression with a radical in the numerator and denominator?

A: To simplify a rational expression with a radical in the numerator and denominator, you need to multiply the numerator and denominator by the conjugate of the radical in the denominator.

Q: Can I simplify a rational expression with a trigonometric function in the numerator and denominator?

A: Yes, you can simplify a rational expression with a trigonometric function in the numerator and denominator, but you need to follow the rules of trigonometric functions.

Q: How do I simplify a rational expression with a logarithmic function in the numerator and denominator?

A: To simplify a rational expression with a logarithmic function in the numerator and denominator, you need to multiply the numerator and denominator by the reciprocal of the logarithmic function in the denominator.

Q: Can I simplify a rational expression with a polynomial in the numerator and denominator?

A: Yes, you can simplify a rational expression with a polynomial in the numerator and denominator, but you need to follow the rules of polynomials.

Q: How do I simplify a rational expression with a rational function in the numerator and denominator?

A: To simplify a rational expression with a rational function in the numerator and denominator, you need to multiply the numerator and denominator by the reciprocal of the rational function in the denominator.

Q: Can I simplify a rational expression with a transcendental function in the numerator and denominator?

A: Yes, you can simplify a rational expression with a transcendental function in the numerator and denominator, but you need to follow the rules of transcendental functions.

Q: How do I simplify a rational expression with a mixed fraction in the numerator and denominator?

A: To simplify a rational expression with a mixed fraction in the numerator and denominator, you need to convert the mixed fraction to an improper fraction and then simplify.

Q: Can I simplify a rational expression with a decimal in the numerator and denominator?

A: Yes, you can simplify a rational expression with a decimal in the numerator and denominator, but you need to follow the rules of decimals.

Q: How do I simplify a rational expression with a percentage in the numerator and denominator?

A: To simplify a rational expression with a percentage in the numerator and denominator, you need to convert the percentage to a decimal and then simplify.

Q: Can I simplify a rational expression with a fraction in the numerator and a variable in the denominator?

A: Yes, you can simplify a rational expression with a fraction in the numerator and a variable in the denominator, but you need to follow the rules of