Sign In
| 12 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
The distance that a vehicle can travel per one gallon of fuel is measured as its mile per gallon (mpg) fuel economy. Each car has two fuel economy numbers, one measuring its efficiency for city driving and the other for highway driving. The combined fuel economy C for x mpg in the city and y mpg on the highway is computed by the following formula. C = 1/12( 1x + 1y )
A rational expression is a fraction where both the numerator and the denominator are polynomials.
p(x)/q(x)
Here, p(x) and q(x) are polynomials and q(x)≠ 0. The expression below is an example of a rational expression. x^2-7/x^3+5x A rational expression is said to be written in its simplest form if the numerator and denominator have no common factors.
Rational Expressions | |
---|---|
Not in Simplest Form | In Simplest Form |
xy/x(x-3)(y+2) | x-1/x+1 |
x^4+x^2/x^2+1 | x^3+7/x^2-x-6 |
Notice that for some of the expressions shown in the table, there are some x-values that make the denominator 0. For example, the denominator of x-1x+1 is 0 when x=- 1. Any value of a variable for which a rational expression is undefined is called an excluded value.
Expression | Restriction | Excluded Value(s) |
---|---|---|
x-1/x+1 | x+1≠ 0 | x ≠ - 1 |
x^3+7/x^2-x-6 | x^2-x-6≠ 0 | x≠ - 2 and x≠ 3 |
xy/x(x-3)(y+2) | x(x-3)(y+2)≠ 0 | x≠0, x ≠ 3, and y ≠ - 2 |
x^4+x^2/x^2+1 | There is no real number that makes x^2+1 zero | None |
Simplifying a rational expression can remove some of the excluded values that appear in the original expression. A rational expression and its simplified form must have the same domain in order for them to be equivalent expressions. This means that the excluded values that are no longer visible in the simplified expression must still be declared.
Equivalent Expressions | |
---|---|
Rational Expression | Simplified Form |
x-3/(x+2)(x-3), x≠ - 2, 3 | 1/x+2,x≠ - 2, 3 |
x^2+2x+1/x^2-1, x≠ - 1, 1 | x+1/x-1,x≠ - 1,1 |
x^3-2x^2+x/x^2,x≠ 0 | x^2-2x+1/x, x≠ 0 |
A rational expression is undefined when its denominator is 0. The values that make the denominator of a rational expression equal to 0 are called excluded values because they are excluded from its domain. Determine the excluded values for the indicated rational expressions.
Split into factors
Factor out x
Write as a power
a^2-b^2=(a+b)(a-b)
Split into factors
Write as a power
a^2-2ab+b^2=(a-b)^2
Commutative Property of Multiplication
a-b=-(b-a)
Before simplifying the common factors, check if there are any restrictions on x. (x - 3)^2/- (x - 3)x(3 + x) Note that the expression is undefined when x=- 3, x=0, or x=3.
Kevin and Zosia are asked to find the values that make the following rational expression undefined. x^2+12x+35/x^2+2x-35 They disagree about the domain of the rational expression.
A rational expression is undefined for values that make its denominator zero. Therefore, those values should be excluded from the domain.
Use the Zero Product Property
(I): LHS-7=RHS-7
(II): LHS+5=RHS+5
Operations with rational numbers and rational expressions are similar.
Multiplying rational expressions works the same way as multiplying fractions. The numerators and denominators are multiplied separately.
P(x)/Q(x) * H(x)/G(x)=P(x)* H(x)/Q(x)* G(x)
Multiply fractions
Cancel out common factors
Simplify quotient
a* a=a^2
Dividing two rational expressions is the same as multiplying the first expression by the reciprocal of the second expression.
P(x)/Q(x) ÷ H(x)/G(x) = P(x)/Q(x) * G(x)/H(x)
To divide a rational expression by another rational expression, multiply the first expression by the reciprocal of the second. 5-x/x^2+3x ÷ x^2-25/x^2+10x+21 [0.5em] ⇕ [0.6em] 5-x/x^2+3x * x^2+10x+21/x^2-25 Once the quotient is expressed as a product, the remaining steps are the same as those for multiplying rational expressions.
Next, the numerators and denominators can be multiplied. 5-x/x(x+3) * (x+3)(x+7)/(x+5)(x-5) [0.5em] ⇕ [0.6em] (5-x)(x+3)(x+7)/x(x+3)(x+5)(x-5) This product is undefined when x= - 5, x=- 3, x=0, and x=5. Also, the values that make the divisor's denominator in the original quotient expression equal 0 should be excluded. These values are x=- 7 and x=- 3. Excluded Values - 7, - 5, - 3, 0, 5
Factor out - 1
Cancel out common factors
Simplify quotient
Distribute - 1
Ramsha drew the plan of her house and labeled the sides, measured in meters, as shown.
A= l w According to the diagram, the length l of the house is represented by 2x^2-6xx^2+18x+81 and the width w by 9x+81x^2-9.
l= 2x^2-6x/x^2+18x+81, w= 9x+81/x^2-9
Factor out 2x
Split into factors
Commutative Property of Multiplication
Write as a power
a^2+2ab+b^2=(a+b)^2
Factor out 9
Write as a power
a^2-b^2=(a+b)(a-b)
Multiply fractions
a^2=a* a
Cancel out common factors
Simplify quotient
Multiply
Denominator | Restrictions on the Denominator | Restrictions on the Variable |
---|---|---|
(x+9)^2 | (x+9)^2≠ 0 | x≠ - 9 |
(x+3)(x-3) | x+3≠ 0 and x-3≠ 0 | x≠ -3 and x≠ 3 |
(x+9)(x+3) | x+9≠ 0 and x+3≠ 0 | x≠ -9 and x≠ -3 |
There are three unique restrictions on the variable x. x≠ - 9, x≠ - 3, x≠ 3
Companies aim to produce packaging using the lowest possible amount of material. They produce their packages in such a way that the ratio of the surface area S of a package to its volume V is as small as possible. Efficiency Ratio: S/V A company is designing two different types of packages. The table shows the expressions for their surface areas and volumes.
Surface Area, S | Volume, V | |
---|---|---|
Type I | x^3-x^2/4x^4+12x^3 | x^2-2x+1/x^3+3x^2 |
Type II | 2x^2-9x-18/4x^2-28x+24 | 2x^2+x-3 |
Factor out x^2
Factor out 4x^3
Factor out x^2
Identity Property of Multiplication
Write as a power
(a-b)^2=a^2-2ab+b^2
Multiply fractions
Write power as a product
Cancel out common factors
Simplify quotient
Multiplying Rational Expressions | |
---|---|
Product | 2x^2-9x-18/4x^2-28x+24 * 1/2x^2+x-3 |
Factor | (2x+3)(x-6)/4(x-1)(x-6) * 1/(2x+3)(x-1) |
Multiply | (2x+3)(x-6)/4(x-1)(x-6)(2x+3)(x-1) |
Cancel Out Common Factors | (2x+3)(x-6)/4(x-1)(x-6)(2x+3)(x-1) |
Simplify | 1/4(x-1)^2 |
The efficiency ratio of Type II is 14(x-1)^2
Type I | Type II | |
---|---|---|
Efficiency Ratio | x/4(x-1) | 1/4(x-1)^2 |
Substitute | 2/4( 2-1) | 1/4( 2-1)^2 |
Evaluate | 1/2 | 1/4 |
Recall that the smaller the ratio, the more efficient the packaging. Therefore, Type II is more efficient because 14 < 12.
A complex fraction is a rational expression where the numerator, denominator, or both, contain a rational expression.
p(x)r(x)/q(x)m(x)
Here, p(x), r(x), q(x), and m(x) are polynomials. A complex fraction can be simplified by rewriting it as a quotient and then dividing the rational expressions. p(x)r(x)/q(x)m(x) ⇔ p(x)/r(x) ÷ q(x)/m(x) As an example of a complex fraction, consider the following division of rational expressions.
x^2-2x+4x^2-y^2/x-yx+y ⇔ x^2-2x+4/x^2-y^2 ÷ x-y/x+yAnimals adapt to their environment. As a result of adaptations, the surface area and volume of animals vary depending on where they live. For example, penguins have a lower surface area to volume ratio to conserve their body heat.
Start by rewriting the complex fraction as a division expression and then divide the rational expressions.
Factor out 2π r^2
Factor out 3
Factor out 3
Split into factors
Commutative Property of Multiplication
Write as a power
(a-b)^2=a^2-2ab+b^2
Factor out π r^2h
Multiply fractions
Write power as a product
Cancel out common factors
Simplify quotient
With the methods seen in the lesson, the challenge given at the beginning can finally be solved. Recall the combined fuel economy formula. C = 1/12 ( 1x+ 1y ) In the formula, x represents miles per gallon (mpg) in the city and y represents miles per gallon on the highway.
x= 28, y= 36
a/b=a * 9/b * 9
a/b=a * 7/b * 7
Add fractions
Multiply fractions
1/a/b= b/a
Calculate quotient
.1 /a/b.=b/a
a/b=a * 10/b * 10
Add fractions
Multiply fractions
.1 /a/b.=b/a
Simplify the rational expression.
We want to simplify the given rational expression. To do so, we will factor the numerator and denominator as much as possible. Then, we will cancel out any common factors. Let's do it!
We simplified the given expression.
Let's factor the numerator and denominator as much as possible. If they have any common factors, we will cancel them out.
We have simplified the expression.
Find the product of the rational expressions.
We want to multiply the rational expressions. 6x^2y^3/x^3 * 3x^2y/10x^3y^2 We multiply numerator by numerator and denominator by denominator. Then, we can cancel out any common factors.
In a similar fashion, we will multiply the given rational expressions. 6xy^3/x^6 * x^4/36xy^4 Let's do it!
Find the quotient of the rational expressions.
We want to divide the rational expressions. 7x+7y/y-x÷ 28/5x-5y We will begin by rewriting the division as a product. To be more specific, we will multiply 7x+7yy-x by the reciprocal of 285x-5y. 7x+7y/y-x * 5x-5y/28 The remaining steps are the same as those for multiplying rational expressions. We can factor the numerators and denominators of each expression and cancel out any common factors. Let's start with the first expression.
Now, let's factor the second expression.
Finally, we can multiply the expressions and cancel out any common factors.
Even if the order of the steps we follow in dividing rational expressions changes, the result will not change. 5x/2x-10÷ x^2+6x/x^2+x-30 We can begin by factoring the numerators and denominators of each expression. Since the numerator of the first expression is already factored, we will factor only the denominator.
Now, let's factor the second expression.
We can now multiply the first expression by the reciprocal of the second expression. 5x/2(x-5) ÷ x(x+6)/(x-5)(x+6) ⇕ 5x/2(x-5) * (x-5)(x+6)/x(x+6) Finally, we cancel out any common factors.
We want to simplify the given rational expression. To do so, we will factor the numerator and denominator as much as possible. Then, we will cancel out any common factors.
The expression cannot be factored any further.
We will identify the restrictions on the variable from the denominator of the simplified expression and from any other denominator used. For simplicity, we will use their factored forms.
Denominator | Restrictions on the Denominator | Restrictions on the Variable |
---|---|---|
(x-5)(3x^2-5) | x-5 ≠ 0 and 3x^2-5 ≠ 0 | x≠ 5 and x = ± sqrt(5/3) |
x-5 | x-5≠ 0 | x≠ 5 |
We see that there are three unique restrictions on the variable x. x ≠ - sqrt(5/3), x ≠ sqrt(5/3), x ≠ 5
A simplified rational expression will not have any common factors in the numerator and denominator. Let's try to look for any common factors in the given rational expressions one at a time.
Let's factor the numerator and denominator.
In option B, we are given a multiplication of two rational expressions. Let's factor the numerators and denominators of each expression before multiplying the rational expressions.
Now we can multiply and simplify common factors.
We need to find the quotient of the rational expressions. Since the divisor is fully factored, we only need to factor the dividend.
Recall that dividing by a rational expression is the same as multiplying by the reciprocal of the rational expression. The reciprocal of x+5x is xx+5.
Finally, we will simplify the given complex fraction.
After simplifying and performing operations, we see that only option B is simplified to a different expression. Therefore, the answer is option B.
Given | Simplified Form | |
---|---|---|
Option A | x^2-x/x^2+2x-3 | x/x+3 |
Option B | x-1/x^2-1 * x^2+4x+3/x | x+3/x |
Option C | x^2+2x-15/x^2-9 ÷ x+5/x | x/x+3 |
Option D | 13y-3/x+33xy-3x | x/x+3 |