Consider The Functions Given Below:${ \begin{array}{l} P(x) = \frac{2}{3x - 1} \ Q(x) = \frac{6}{-3x + 2} \end{array} } M A T C H E A C H E X P R E S S I O N W I T H I T S S I M P L I F I E D F O R M . N O T A L L E X P R E S S I O N S W I L L B E U S E D .1. \[ Match Each Expression With Its Simplified Form. Not All Expressions Will Be Used.1. \[ M A T C H E A C H E X P Ress I O N W I T Hi T Ss Im Pl I F I E Df Or M . N O T A Ll E X P Ress I O N S W I Ll B E U Se D .1. \[ \frac{-3x + 2}{3(3x -

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 functions P(x) and Q(x) and match each expression with its simplified form. We will also provide a step-by-step guide on how to simplify rational expressions.

Understanding Rational Expressions

A rational expression is a fraction that contains variables and constants in the numerator and denominator. Rational expressions can be simplified by canceling out common factors in the numerator and denominator.

The Functions P(x) and Q(x)

The functions P(x) and Q(x) are given by:

${ P(x) = \frac{2}{3x - 1} }$ ${ Q(x) = \frac{6}{-3x + 2} }$

Simplifying P(x)

To simplify P(x), we need to find the common factors in the numerator and denominator. In this case, there are no common factors, so we cannot simplify P(x) further.

Simplifying Q(x)

To simplify Q(x), we need to find the common factors in the numerator and denominator. In this case, we can factor out a -3 from the numerator and a -3 from the denominator.

${ Q(x) = \frac{6}{-3x + 2} }$ ${ Q(x) = \frac{-2 \cdot 3}{-3(x - \frac{2}{3})} }$ ${ Q(x) = \frac{-2}{x - \frac{2}{3}} }$

Matching Expressions with Simplified Forms

Now that we have simplified P(x) and Q(x), we can match each expression with its simplified form.

1. ${ \frac{-3x + 2}{3(3x - 1)} }$ This expression can be simplified by canceling out the common factor of 3 in the numerator and denominator.

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Q: What is a rational expression?

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

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to find the common factors in the numerator and denominator and cancel them out.

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

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

• Not factoring the numerator and denominator
• Not canceling out common factors
• Not simplifying the expression further

Q: How do I factor the numerator and denominator of a rational expression?

A: To factor the numerator and denominator of a rational expression, you need to find the greatest common factor (GCF) of the terms and factor it out.

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

A: The greatest common factor (GCF) is the largest factor that divides all the terms in a set of numbers.

Q: How do I cancel out common factors in a rational expression?

A: To cancel out common factors in a rational expression, you need to divide the numerator and denominator by the common factor.

Q: What is the difference between simplifying and canceling out common factors?

A: Simplifying a rational expression involves finding the greatest common factor (GCF) of the numerator and denominator and dividing both by the GCF. Canceling out common factors involves dividing the numerator and denominator by a common factor, but not necessarily the GCF.

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

A: Yes, you can simplify a rational expression that has a variable in the denominator. However, you need to be careful not to divide by zero.

Q: How do I handle a rational expression with a variable in the denominator that is equal to zero?

A: If a rational expression has a variable in the denominator that is equal to zero, you need to check if the expression is undefined. If it is, you need to simplify the expression further or use a different method to solve the problem.

Q: Can I simplify a rational expression that has a negative exponent?

A: Yes, you can simplify a rational expression that has a negative exponent. To do this, you need to rewrite the expression with a positive exponent and then simplify.

Q: How do I simplify a rational expression that has a negative exponent?

A: To simplify a rational expression that has a negative exponent, you need to rewrite the expression with a positive exponent. This can be done by flipping the fraction and changing the sign of the exponent.

Q: Can I simplify a rational expression that has a fraction in the denominator?

A: Yes, you can simplify a rational expression that has a fraction in the denominator. To do this, you need to multiply the numerator and denominator by the reciprocal of the fraction in the denominator.

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

A: To simplify a rational expression that has a fraction in the denominator, you need to multiply the numerator and denominator by the reciprocal of the fraction in the denominator. This will eliminate the fraction in the denominator and allow you to simplify the expression further.

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

A: Yes, you can simplify a rational expression that has a variable in the numerator and denominator. To do this, you need to find the greatest common factor (GCF) of the terms and factor it out.

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

A: To simplify a rational expression that has a variable in the numerator and denominator, you need to find the greatest common factor (GCF) of the terms and factor it out. This will allow you to cancel out common factors and simplify the expression further.

Q: Can I simplify a rational expression that has a complex fraction?

A: Yes, you can simplify a rational expression that has a complex fraction. To do this, you need to multiply the numerator and denominator by the reciprocal of the fraction in the denominator.

Q: How do I simplify a rational expression that has a complex fraction?

A: To simplify a rational expression that has a complex fraction, you need to multiply the numerator and denominator by the reciprocal of the fraction in the denominator. This will eliminate the fraction in the denominator and allow you to simplify the expression further.