Day: May 20, 2018

Week 14 – Rational Expressions

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This week in Pre-Calculus 11 we started the rational expressions unit. This week we learned about equivalent rational expressions and how to multiply and divide rational expressions.

What is a Rational Expression: A rational expression is a number that can be expressed as a fraction. For example, the number 3 can be represented as \frac{3}{1}, therefore it is a rational number. Irrational numbers are numbers that form an endless amount of decimals and are non-repeating.

Non Permissible Values: When there is a variable in the denominator of a fraction you have to give it a restriction. This is an important step because the denominator of a fraction can never be zero because that would make the fraction undefined.

Ex. \frac{1}{x-4} 

x-4\neq0

x\neq4

In the example above, because there was a variable in the denominator I had to create a restriction. The first step is to make the denominator equal to zero. Then, I isolated the variable to figure out a value that would make the fraction undefined.

What are Equivalent Rational Expressions? Equivalent rational expressions are fractions where the numerator and denominator have something in common, so that they can simplify.

Ex. \frac{10}{25} 

\frac{10}{25}=\frac{5\times2}{5\times5}

\frac{10}{25}=\frac{2}{5}

In the example above, I simplified a fraction by taking something out that both the numerator and denominator had in common. I took something in common out by factoring the fraction first, then cancelling out equal values. Similarly, in fractions that involve variables you have to factor the numerator and denominator first and then take out common terms.

Ex. \frac{3x-12}{x^2+x-20} 

\frac{3(x-4)}{(x+5)(x-4)} 

\frac{3}{x+5} 

x-4\neq0

x\neq4

x+5\neq0

x\neq-5 

In the example, the first thing I did was factor the numerator and denominator. Then, I took out the factors that they had in common. After, I figured out the non permissible values for the original expression and the factored expression.

How To Multiply and Divide Rational Expressions: To multiply and divide rational expressions you just have to multiply straight across. But it is important to factor first because it will be easier to simply the expression later.

Ex. \frac{2x+2}{3x}\times\frac{x^2-2x}{4x+4}

\frac{2(x+1)}{3x}\times\frac{x(x-2)}{4(x+1)}

\frac{2(x+1)(x)(x-2)}{3x(4)(x+1)}

\frac{2x(x-2)}{3x(4)}

\frac{2x(x-2)}{12x}

\frac{x-2}{6}

3x\neq0

x\neq0

x-2\neq0

x\neq2

The first thing I did was factor both of the rational expressions and then I took out the factors that were the same. After that, I simplified the fraction because 2 and 12 share a common factor of 2. Then, I found the non permissible values for the original and factored rational expression.

How To Divide Rational Expressions: To divide rational expressions you do the exact same as multiplying, by factoring and then taking out common factors. But when dividing fractions you have to remember to multiply by the reciprocal of the second fraction. You also have to find the non permissible values for the variable.

Ex. \frac{x+5}{x-4}\div\frac{x^2-25}{3x-12} 

\frac{x+5}{(x-4)}\div\frac{(x-5)(x+5)}{3(x-4)}

\frac{x+5}{(x-4)}\times\frac{3(x-4)}{(x-5)(x+5)}

\frac{(x+5)(3)(x-4)}{(x-4)(x-5)(x+5)}

\frac{3}{x-5} 

x-4\neq0

x\neq4

x+5\neq0

x\neq-5

x-5\neq0

x\neq5

For dividing the rational expression, the first step I did was to factor both of the fractions. Then, I reciprocated the second fraction and multiplied them together. After, I took out the common factors and everything that was left over become the new rational expression.