Section 6.1 Rational Expressions

OBJECTIVES A Find the numbers that make a rational expression undefined.

OBJECTIVES B Write an equivalent fraction with the indicated denominator.

OBJECTIVES C Write a fraction in the standard forms.

OBJECTIVES D Reduce a fraction to lowest terms.

DEFINITION If P and Q are polynomials: Rational Expressions

DEFINITION The variables in a rational expression may not be replaced by values that will make the denominator zero. Undefined Rational Expressions

DEFINITION If P, Q, and K are polynomials Fundamental Property of Fractions

Reducing Fractions PROCEDURE 1.Write numerator and denominator in factored form. 2. Find the GCF.

Reducing Fractions PROCEDURE 3.Replace the quotient of the common factors by Rewrite in lowest terms.

DEFINITION Quotient of Additive Inverses

Practice Test Exercise #1 Chapter 6 Section 6.1A,B

Find the undefined value(s) for

Write the fraction with the indicated denominator.

Practice Test Exercise #2 Chapter 6 Section 6.1C

Write in standard form

Practice Test Exercise #4 Chapter 6 Section 6.1D

Reduce to lowest terms. Factor out – 1 Difference of Squares Difference of Cubes

Reduce to lowest terms.

Section 6.2 Multiplication and Division of Rational Expressions

OBJECTIVES A Multiply rational expressions.

OBJECTIVES B Divide rational expressions.

OBJECTIVES C Use multiplication and division together.

DEFINITION Multiplication of Rational Expressions

To Multiply Rational Expressions PROCEDURE 1.Factor the numerators and denominators completely. 2. Simplify each expression.

To Multiply Rational Expressions PROCEDURE 3.Multiply remaining factors. 4.The final product should be in lowest terms.

DEFINITION Division of Real Numbers

Practice Test Exercise #6 Chapter 6 Section 6.2B

Perform the indicated operations.

Practice Test Exercise #7 Chapter 6 Section 6.2C

Perform the indicated operations.

Section 6.3 Addition and Subtraction of Rational Expressions

OBJECTIVES A Add or subtract rational expressions with the same denominator.

OBJECTIVES B Add or subtract rational expressions with different denominators.

Finding the LCD of Two or More Rational Expressions PROCEDURE 1.Factor denominators. Place factors in columns. ( Not necessary to factor monomials ).

Finding the LCD of Two or More Rational Expressions PROCEDURE 2.Select the factor with the greatest exponent from each column.

Finding the LCD of Two or More Rational Expressions PROCEDURE 3.The product of all the factors obtained is the LCD.

To Add or Subtract Fractions with Different Denominators. PROCEDURE 1.Find the LCD. 2.Write all fractions as equivalent ones with LCD as denominator.

To Add or Subtract Fractions with Different Denominators. PROCEDURE 3.Add numerators. 4.Simplify.

Practice Test Exercise #9a Chapter 6 Section 6.3B

Perform the indicated operations.

Section 6.4 Complex Fractions

OBJECTIVES A Write a complex fraction as a simple fraction in reduced form.

Simplifying Complex Fractions PROCEDURE Multiply the numerator and denominator of the complex fraction by the LCD of all simple fractions. METHOD 1

PROCEDURE Perform operations indicated in numerator and denominator. Then divide numerator by denominator. Simplifying Complex Fractions METHOD 2

Practice Test Exercise #10 Chapter 6 Section 6.4A

Simplify. Multiply by LCD

Simplify.

Section 6.5 Division of Polynomials and Synthetic Division

OBJECTIVES A Divide a polynomial by a monomial.

OBJECTIVES B Use long division to divide one polynomial by another.

OBJECTIVES C Completely factor a polynomial when one of the factors is known.

OBJECTIVES D Use synthetic division to divide one polynomial by a binomial.

OBJECTIVES E Use the remainder theorem to verify that a number is a solution of a given equation.

Dividing a Polynomial by a Monomial RULE Divide each term in the polynomial by the monomial.

DEFINITION The Remainder Theorem

DEFINITION The Factor Theorem

Practice Test Exercise #13 Chapter 6 Section 6.5B

Divide. Write in descending order.

Divide. Remainder

Practice Test Exercise #14 Chapter 6 Section 6.5C

0

Practice Test Exercise #16 Chapter 6 Section 6.5E

–1 1–4 –

Section 6.6 Equations Involving Rational Expressions

OBJECTIVES A Solve equations involving rational expressions.

OBJECTIVES B Solve applications using proportions.

Solving Equations Containing Rational Expressions PROCEDURE 1.Factor denominators and multiply both sides of the equation by the LCD.

PROCEDURE 2.Write the result in reduced form. Use the distributive property to remove parentheses. Solving Equations Containing Rational Expressions

PROCEDURE 3.Determine whether the equation is linear or quadratic and solve accordingly. Solving Equations Containing Rational Expressions

PROCEDURE 4.Check that the proposed solution satisfies the equation. If not, discard it as an extraneous solution. Solving Equations Containing Rational Expressions

DEFINITION Property of Proportions A proportion is true if the cross products are equal.

Practice Test Exercise #18 Chapter 6 Section 6.6A

Solve:

O F F

Practice Test Exercise #19 Chapter 6 Section 6.6B

a.

Section 6.7 Applications: Problem Solving

OBJECTIVES A Solve integer problems.

OBJECTIVES B Solve work problems.

OBJECTIVES C Solve distance problems.

OBJECTIVES D Solve for a specified variable.

PROCEDURE: Read Select Think Use Verify RSTUV Method for Solving Word Problems

Practice Test Exercise #21 Chapter 6 Section 6.7B

Section 6.8 Variation

OBJECTIVES A Direct variation.

OBJECTIVES B Inverse variation.

OBJECTIVES C Joint variation.

OBJECTIVES D Solve applications involving direct, inverse, and joint variation.

DEFINITION Direct Variation y varies directly as x if there is a constant k :

DEFINITION Inverse Variation y varies inversely as x if there is a constant k :

DEFINITION Joint Variation z varies jointly with x and y if there is a constant k :

Practice Test Exercise #24 Chapter 6 Section 6.8A