Tuesday, January 24, 2023

# Simplifying Rational Expressions Practice Algebra 2

## Simplifying Complex Rational Expressions

Simplifying rational expressions introduction | Algebra II | Khan Academy

A complex rational expression is a rational expression that contains additional rational expressions in the numerator, the denominator, or both. We can simplify complex rational expressions by rewriting the numerator and denominator as single rational expressions and dividing. The complex rational expression a can be simplified by rewriting the numerator as the fraction a and combining the expressions in the denominator as 1 b . We can then rewrite the expression as a multiplication problem using the reciprocal of the denominator. We get a

Given a complex rational expression, simplify it.

• Combine the expressions in the numerator into a single rational expression by adding or subtracting.
• Combine the expressions in the denominator into a single rational expression by adding or subtracting.
• Rewrite as the numerator divided by the denominator.
• Rewrite as multiplication.
• Begin by combining the expressions in the numerator into one expression.

Now the numerator is a single rational expression and the denominator is a single rational expression.

We can rewrite this as division, and then multiplication.

Can a complex rational expression always be simplified?

Yes. We can always rewrite a complex rational expression as a simplified rational expression.

## Simplify Rational Expressions With Opposite Factors

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. Letâs start with a numerical fraction, say 7 . We know this fraction simplifies to â1 . We also recognize that the numerator and denominator are opposites.

In Foundations, we introduced opposite notation: the opposite of a

, whose numerator and denominator are opposites, in this way:

So, in the same way, we can simplify the fraction x

In general, we could write the opposite of a

## Solve Direct Variation Problems

In the following exercises, solve.

89. If \ varies directly as \ when \ and \, find \ when \.

90. If \ varies inversely as \ when \ and \ find \ when \.

91. Vanessa is traveling to see her fiancÃ©. The distance, \, varies directly with the speed, \, she drives. If she travels 258 miles driving 60 mph, how far would she travel going 70 mph?

301 mph

## Simplifying Rational Expressions Explanation & Examples

Now that you understand what rational numbers are, the next topic to look at in this article is rational expressions and how to simplify them. Just for your own benefit, we define a rational number as a number expressed in the form of p/q where it is not equal to zero.

In other words, we can say a rational number is nothing more than a fraction in which the numerator and the denominator are integers. Examples of rational numbers are 5/7, 4/9/ 1/ 2, 0/3, 0/6 etc.

On the other hand, a rational expression is an algebraic expression of the form f / g in which the numerator or denominator are polynomials, or both the numerator and the numerator are polynomials.

Examples of rational expression are 5/x 2, 4/, /5, /, /, /2x etc.

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## Multiply And Divide Rational Functions

19. Find \=f \cdot g\) where \=\dfrac+9 x}-3 x-4}\) and \=\dfrac-16}+12 x}\).

\=3\)

20. Find \=\dfrac\) where \=\dfrac}\) and \=\dfrac+54 x}-x-42} \).

Add and Subtract Rational Expressions with a Common Denominator

In the following exercises, perform the indicated operations.

21. \

22. \

23. \

24. \

25. \

26. \

27. \

28. \

## How To

• Step 1. Factor the numerator and denominator completely.
• Step 2. Simplify by dividing out common factors.
• Usually, we leave the simplified rational expression in factored form. This way it is easy to check that we have removed all the common factors!

Weâll use the methods we covered in Factoring to factor the polynomials in the numerators and denominators in the following examples.

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