Chapter 6
solving rational equations
1. factor all denominators 2. determine which numbers cannot be solutions of the equation 3. multiply both sides of the equation by the LCD of all rational expressions in the equation 4. use the distributive property to remove parentheses, remove any factors equal to 1, and write the result in simplified form 5. solve the resulting equation 6. check all possible solutions in the original equation
finding the LCD
1. factor each denominator completely 2. the LCD is a product that uses each different factor obtained in step 1 the greatest number of times it appears in any one factorization
simplifying rational expressions
1. factor the numerator and denominator completely to determine their common factors 2. remove factor equal to 1 by replacing each pair of factors common to the numerator and denominator with the equivalent fraction 1/1
adding and subtracting rational expressions with unlike denominators
1. find LCD 2. write each rational expression as an equivalent expression whose denominator is the LCD 3. add or subtract the numerators and write the sum or difference over the LCD 4. simplify the resulting rational expression, if possible
multiplying rational expressions
Let A, B, C, and D represent polynomials, where B and D are not 0, Then simplify, if possible
methods for simplifying complex fractions
METHOD 1 1. write the numerator and the denominator of the complex fraction as a single rational expression 2. perform the division by multiplying the numerator of the complex fraction by the reciprocal of the denominator METHOD 2 1. find the LCD of all rational expressions in the complex fraction 2. multiply the complex fraction by 1 in the form LCD/LCD 3. perform the operations in the numerator and denominator. no fractional expressions should remain within the complex fraction 4. simplify the result, if possible
algorithm
a repeating series of steps...used when the divisor is not a monomial
terms, extremes, means
a/b = c/d ...the terms a and d are called the extremes of the proportion, and the terms b and c are called the means
proportion
an equation indicating that 2 ratios or rates are equal
rational expressions
an expression of the form P/Q, where P and Q are polynomials and Q does not equal 0
rational functions
function whose equation is defined by a rational expression in one variable, where the value of the polynomial in the denominator is never 0
similar triangles
if 2 triangles are similar, then 1. the 3 angles of the first triangle have the same measure, respectively, as the three angles of the second triangle 2. the lengths of all corresponding sides are in proportion
adding and subtracting rational expressions
if A/D and B/D are rational expressions, Then simplify, if possible
factor theorem
if P(x) is a polynomial in x, then P(k) = 0 iff x-k is a factor of P(x)
continued fraction
if a fraction has a complex fraction in its numerator or denominator
rate of work
if a job can be completed in x house, the rate of work can be expressed as: 1/x of the job is completed per hour. if a job is completed in some other unit of time, such as x minutes or x days, then the rate of work is expressed in that unit
remainder theorem
if a polynomial P(x) is divided by x-k, the remainder is P(k)
properties of fractions
if a, b, c, d, and k represent real numbers, and if there are no divisions by 0, then.... 1. 2. 3. 4.
rational equation
if an equation contains one or more rational expressions
joint variation
if one variable varies directly with the product of two or more variables, the relationship is called joint variation. if y varies jointly with x and z, then y = kxz. the nonzero constant k is called the constant of variation
opposites
if the terms of two polynomials are the same, except that they are opposite in sign, the polynomials are opposites
the fundamental property of proportions
in a proportion, the product of the extremes is equal to the product of the means. If a/b = c/d, then ad = bc and if ad = bc, then a/b = c/d
dividing rational expressions
let A, B, C, and D represent polynomials, where B, C, and D are not 0, Then simplify, if possible
combined variation
many applied problems involve a combination of direct and inverse variation
complex rational expression, or more simply, a complex fraction
rational expression whose numerator and/or denominator contain rational expressions
least common denominator (LCD)
terms of the smallest common denominator possible
the quotient of opposites
the quotient of any nonzero polynomial and its opposite is -1
ratio
the quotient of two numbers or two quantities with the same units
direct variate
the words "y varies directly with x" or "y is directly proportional to x" means that y=kx for some nonzero constant k. the constant k is called the constant of variation or the constant of proportionality
inverse variation
the words "y varies inversely with x" or "y is inversely proportional to x" means that y= k/x for some nonzero constant k. the constant k is called the constant of variation
building rational expressions
to build a rational expression, multiply it by 1 in the form c/c, where c is any nonzero number or expression
dividing a polynomial by a monomial
to divide a polynomial by a monomial, divide each term of the polynomial by the monomial Let a, b, and d represent monomials, where d is not 0
solving variation problems
to solve a variation problem 1. translate the verbal model into an equation 2. substitute the first set of values into the equation from step 1 to determine the value of k 3. substitute the value of k into the equation from step 1 4. substitute the remaining set of values into the equation from step 3 and solve for the unknown
synthetic division
used to divide a polynomial by a binomial of the form x-k
asymptote
when a graph approaches a line, we call the line an asymptote
multiplying by -1
when a polynomial is multiplied by -1, the result is its opposite
rate
when we compare two quantities having different units, we call the comparison a rate, and we can write it as a fraction
extraneous solutions
when we multiply both sides of an equation by a quantity that contains a variable, we can get false solutions.