Rational Expressions - Gavilan College -> Faculty / …hhh.gavilan.edu/ybutterworth/algebra/ch6EMG.doc · Web viewSS6.4 Adding & Subtracting Rational Expressions w/ Unlike Denominators

SS6.1 Simplifying Rational Expressions

Recall that a rational number is a quotient of integers. A rational expression is a quotient of polynomials, such as:

P where Q0 and P and Q are polynomials Q

A rational expression can be evaluated just as any polynomial, except that a rational expression can be undefined when the denominator is equal to zero.

Example: Evaluate x 2 3 when x = 5 x + 9

Example: For what number(s) is x 2 3 undefined x2 3x 10

1) Find zeros of the polynomial in the denominator, by solving as a quadratic

Example: Are there any zeros for x 2 + 3x 10 ? x2 + 5

Example: Evaluate x 2 3 when x = -9

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x + 9

Recall that -a = a = _ a when b0 b -b b

This is also true of polynomials. My suggestion is to simply pull any negative signs out and consider the entire expression as either a positive or negative expression.

Example: - ( x + 3 ) = _ x + 3 x2 3x + 2 x2 3x + 2

Just as when we were dealing with fractions, if you multiply the numerator and denominator by the same thing the resulting expression is equivalent. This is called the Fundamental Principle of Rational Expressions, when we are discussing a fraction of polynomials.

PR = P if P, Q and R are polynomials andQR Q Q&R0

Concept Example: 15 = 3 5 = 3 35 7 5 7

In order to simplify rational expressions we will use the Fundamental Principle of Rational Expressions just as we used the Fundamental Principle of Fractions to simplify fractions.

Steps to Simplifying a Rational Expression1) Factor the numerator and the denominator completely

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2) Cancel common factors

Example: Simplify 15x 2 y 35xy2

Recall that another way of using the principle is division!

Example: Simplify x 2 x2 + 2x

Example: Simplify x + 5 x2 + 2x 15

Example: Simplify x 2 x 2 x2 + 5x 14

Example: Simplify - x + y x y

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Your TurnExample: Simplify x 3 + 8

x2 5x 14

Example: Simplify - x 2 3x 2 x2 x 2

SS6.2 Multiplying and Dividing Rational Expressions

Steps for Multiplying Rational Expressions

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1) Factor numerators and denominators completely2) Cancel if possible3) Multiply numerators and denominators

Concept Example: 2 3 = 1 3 8 4

Example: x + 2 x + 1 x2 + 7x + 6 x3 + 8

Example: 7x 2 -3y 15y3 14x3

Your TurnExample: x 2 5x 14 x 2 8x + 7

x3 1 x2 + x 2

Example: -7x -6 y 4 24y3 15x5

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Steps for Dividing Rational Expressions1) Take the reciprocal of the divisor.2) Multiply the dividend and the reciprocal of the divisor

Concept Example: 1 2 = 1 3 = 3 2 3 2 2 4

Example: x + 1 3x 2 3 2x2 x + 1 x2 3 x + 2

Example: 4x 2 16 8x 3 64 x + 1 x2 + 9x + 8

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Your TurnExample: x 2 + 8 x 3 10

x2 16 x2 9x + 20

SS6.3 Adding and Subtracting Rational Expressions w/ Common Denominators

When we have a common denominator, as with fractions, we simply add or subtract the numerators. We do have to be cautious because when

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subtracting it is the entire numerator that's subtracted, so we must use the distributive property to subtract.

Concept Example: 7 3 = 4 15 15 15

Example: 2y + 3 = 7n 7n

Example: 2x + 5 x 5 = y + 2 y + 2

Just as with fractions sometimes we will need to reduce our answer before we are finished.

Concept Example: 2 + 4 = 6 = 2 15 15 15 5

Example: y 2 = y 2 y 2

Your TurnExample: x 2 + 1 + 3x =

x + 2 x + 2

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This leads us to the point that we always need to factor the numerator and denominator in order to reduce and also for our next application of finding a common denominator.

Finding the LCD of a rational expression is the same as finding the LCD of fractions. We just need to remember that a polynomial must first be factored. Each factor that isn't a constant is considered like a prime number.

Concept Example: Find the LCD of 2 , 7 15 36

Example: Find the LCD of 1 , 2x + 5 x x(x + 5)

Example: Find the LCD of x + 2 , x + 1 x2 + 3x + 2 x + 2

1) Factor x2 + 3x + 3

2) LCDExample: Find LCD of 2 , x + 1

4x2 + 13x + 3 x2 + 6x + 9

1) Factor 4x2 + 13x + 3

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2) Factor x2 + 6x + 9

3) LCD

Your TurnExample: Find LCD of 2x , x 2 + 1

x2 + 2x + 1 x2 1

Next, we need to build a higher term so that we can add and subtract rational expressions with unlike denominators.

Building A Higher Term1) Find LCD2) Divide LCD by denominator3) Multiply numerator by quotient

Concept Example: Build the higher term 2 , 7 15 36

1) LCD was 180

2) 180 15= 180 36 =

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3) 2 = 7 = 15 180 36 180

Example: Find the LCD of 1 , 2x + 5 x x(x + 5)

1) LCD was x(x + 5)

2) x(x + 5) = x(x + 5) =x x(x + 5)

3) 1 = 2x + 5 x x(x + 5) x(x + 5)

Your TurnExample: Find LCD of 2 , x + 1

4x2 + 13x + 3 x2 + 6x + 9

1) LCD was (4x + 1)(x + 3)2

2) (4x + 1)(x + 3) 2 = (4x + 1)(x + 3) 2 = (4x + 1)(x + 3) (x + 3)2

3) 2 = (4x + 1)(x + 3) (4x + 1)(x + 3)2=

x + 1 = (x + 3)2 (4x + 1)(x + 3)2

SS6.4 Adding & Subtracting Rational Expressions w/ Unlike Denominators

Addition and Subtraction of Rational Expressions w/ Unlike Denominators1) Find LCD2) Build Higher Term3) Add or Subtract as normal

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Example: 15a + 6b b 5

1) LCD

2) Build Higher Term

3) Add

Example: 15 + x 2x 4 x2 4

1) LCD

2) Build Higher Term

3) Add

Example: x 2 x2 4 4 x2

1) LCD

2) Build Higher Term

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3) Subtract

Example: y + 2 2 y + 3

1) LCD

2) Build the higher term

3) Subtract

Example: x 2 x2 1 x2 2x + 1

1) LCD

2) Build Higher Term

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3) Subtract

Your TurnExample: 2x + x 2 + 1

x2 + 2x + 1 x2 1

SS6.5 Simplifying Complex Fractions

A complex fraction is a fraction with an expression in the numerator and an expression in the denominator.

Simplifying Complex Fractions Method #11) Solve or simplify the problem in the numerator2) Solve or simplify the problem in the denominator

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3) Divide the numerator by the denominator4) Reduce

Example: 3 4

----------------- 2 3

Example: 6x 3 5x2

------------------------- 2x 1 10x

Example: 1 + 2 2 3

------------------------- 5 5 9 6

1) Solve numerator 1 + 2 2 3

2) Solve denominator 5 5 9 6

3) Divide & Reduce

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Example: 1 1 5 x

------------------------- 7 + 1 10 x2

1) Simplify numerator 1 1 5 x

2) Simplify denominator 7 + 1 10 x2

3) Divide numerator by denominator

4) Reduce

Simplifying Complex Fractions Method #21) Find the LCD of the numerator and the denominator fractions2) Multiply numerator and denominator by LCD3) Simplify resulting fraction

Example: 3 4

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----------------- 2 3

1) LCD of 3 and 4

2) Multiply 3 4

----------------- 2 3

3) Simplify

Example: 6x 3 5x2

------------------------- 2x 1 10x

1) LCD of 5x2 and 10x

2) Multiply 6x 3 5x2

------------------------- 2x 1 10x

3) Simplify

Example: 1 + 2 2 3

------------------------- 5 5 9 6

1) Find LCD of 2,3, 9, 6

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2) Multiply 1 + 2 2 3

------------------------- 5 5 9 6

3) Simplify

Example: 1 1 5 x

------------------------- 7 + 1 10 x2

1) Find LCD of 5, x, 10 and x2

2) Multiply 1 1 5 x

------------------------- 7 + 1 10 x2

3) SimplifySS6.6 Solving Equations Containing Rational Expressions

The only difference between solving rational expression equations and regular equations is that all solutions must be checked to make sure that it does not make the denominator of the original expression zero. If one of the solutions makes the denominator zero, it is called an extraneous solution.

Steps1) Find the LCD of the denominators in the equation2) Multiply both sides by the LCD

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3) Solve as usual4) Check for extraneous solutions

Example: 9 = -2 2a 5

1) Find LCD

2) Multiply

3) Solve

4) Check for extraneous solutions 9 = -2 2a 5

Example: 7 + 1 = x x 2 x + 2

1) Find LCD

2) Multiply

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3) Solve

4) Check for extraneous solutions

Examples: y + 2y 16 = 2y 3 2y + 2 4y + 4 y + 1

1) Find LCD

2) Multiply

3) Solve

4) Check for extraneous solutions

Example: 2x 2 = x 8 x + 2 x 2

1) Find LCD

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2) Multiply

3) Solve

4) Check for extraneous solutions

In the next examples, we will want to solve for a variable when there are two variables in an equation.Example: Solve for b: 3a + 2 = -4

3b 2 2a

1) Find LCD

2) Multiply

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3) Solve for b

Example: Solve for n m 2 n = p 6 3 2

1) Find LCD

2) Multiply

3) Solve for n

SS6.7 Ratios and Proportions

A ratio is a quotient of two numbers where the divisor isn't zero. A ratio is stated as: a to b

a : b or a where a & b are whole numbers and b0 b

A proportion is a mathematical statement that two ratios are equal.

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2 = 4 is a proportion 3 6

It is read as: 2 is to 3 as 4 is to 6

The numbers on the diagonal from left to right, 2 and 6, are called the extremes

The numbers on the diagonal from right to left, 4 and 3, are called the means

If we have a true proportion, then the product of the means equals the product of the extremes.

These are also called the cross products and finding the product of the means and extremes is called cross multiplying.

Example: Find the cross products of the following to show that this is a true proportion

27 = 3 72 8

The idea that in a true proportion, the cross products are equal is used to solve for unknowns!

This is strictly a review of material covered as a supplement to Chapter 1. The only problem type that was not covered at that time was the following:

Example: x + 1 = 2 3 3x

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SS6.8 Rational Expressions and Problem Solving

This section is on word problems. In order to do well with word problems you must:

1) Read the problem and understand what is being said and asked2) Write down all the information that is important3) Assign a variable to an unknown and put all other unknowns in

terms of this one4) Translate any equations given in the problem5) Create your final algebraic expression

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6) Solve7) Check

In this section, the only catch is that we are working in parts of things, the distance formula where time is the key or with similar triangles. Therefore, we will have rational expressions coming into our equation. Example: Three times the reciprocal of a number equals 9 times the reciprocal of 6. Find the number.

Example: In 2 minutes, a conveyor belt moves 300 pounds of tin from the delivery truck to a storage area. A smaller belt moves the same amount the same distance in 6 minutes. If both belts are used, find how long it takes to move the cans to the storage area.

Minutes Part of Job in 1 MinuteLargeSmall

Together

Example: A cyclist rode the first 20-mile portion of his workout at a constant speed. For the 16-mile cooldown portion of his workout, he reduced his speed by 2 miles per hour. Each portion of the workout took the same time. Find the cyclist's speed during the first portion and find his speed during the cooldown portion.

Distance = Rate Time

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Similar Triangles are triangles that have proportional side lengths and the same measure in angles

A D

BE

FC

AB : DE as BC : EF

If we don't know the length of a side of a triangle and we know that the triangles are similar, then we can use proportions to solve the problem. The same is true with any figures that are said to be similar.

Example: See #13 on p. 396

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Example: See #35 on p. 396

In this section there are four patterns in the problems as shown by the above examples. Look for these patterns and solve all problems that follow the pattern just as I have solved these problems.

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