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Advanced Algebra
4: Polynomial and Rational Functions

4: Polynomial and Rational Functions

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Oct 6, 2021
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- 3.E: Solving Linear Systems
- 4.1: Algebra of Functions

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4.1: Algebra of Functions
Any polynomial with one variable is a function. The degree of the polynomial is the largest exponent of all the terms. Use function notation to streamline the evaluating process. Substitute the value or expression inside the parentheses for each instance of the variable.
4.2: Factoring Polynomials
The process of writing a number or expression as a product is called factoring5. A useful step in this process is finding the greatest common monomial factor (GCF) of two or more monomials. The GCF of the monomials is the product of the common variable factors and the GCF of the coefficients.
4.3: Factoring Trinomials
Some trinomials of the form x²+bx+c can be factored as a product of binomials. If the trinomial has a greatest common factor, then it is a best practice to first factor out the GCF before attempting to factor it into a product of binomials. If the leading coefficient of a trinomial is negative, then it is a best practice to first factor that negative factor out before attempting to factor the trinomial.
4.4: Solve Polynomial Equations by Factoring
We have learned various techniques for factoring polynomials with up to four terms. The challenge is to identify the type of polynomial and then decide which method to apply.
4.5: Rational Functions - Multiplication and Division
Rational functions have the form r(x)= p(x)/q(x) , where p(x) and q(x) are polynomials and q(x)≠0. Simplifying rational expressions is similar to simplifying fractions.
4.6: Rational Functions - Addition and Subtraction
Adding and subtracting rational expressions is similar to adding and subtracting fractions. Recall that if the denominators are the same, we can add or subtract the numerators and write the result over the common denominator. When working with rational expressions, the common denominator will be a polynomial.
4.7: Solving Rational Equations
A rational equation is an equation containing at least one rational expression. Rational expressions typically contain a variable in the denominator. For this reason, we will take care to ensure that the denominator is not 0 by making note of restrictions and checking our solutions. Solving rational equations involves clearing fractions by multiplying both sides of the equation by the least common denominator (LCD).
4.8: Applications and Variation
Three applications of polynomials and rational functions are discussed: (1) Uniform motion problems involving the formula D=rt , where the distance D is given as the product of the average rate r and the time t traveled at that rate. (2) Work-rate problems that involves multiplying the individual work rate by the time to obtain the portion of the task completed. (3) Variation problems that usually requires multiple steps to solve.
4.E: Polynomial and Rational Functions (Exercises)

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