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.