Chapter 7 Rational Expressions, Equations, and Functions

Chapter 7 Rational Expressions, Equations, and Functions

Section 7.1: Simplifying, Multiplying, and Dividing Rational Expressions and Functions Section 7.2: Adding and Subtracting Rational Expressions Section 7.3: Simplifying Complex Fractions Section 7.4: Solving Rational Equations Putting It All Together

Section 7.1: Simplifying, Multiplying, and Dividing Rational Expressions and Functions Rational Expression A rational expression is an expression of the form P Q where P and Q are polynomials and where Q 0. Note: A fraction (rational expression) equals zero when its numerator equals zero. A fraction (rational expression) is undefined when its denominator equals zero.

Some functions are described by rational expressions. Such functions are called rational functions. f(x) = x + 3 x + 3 is an example of a rational function since is a rational x 8 x 8 expression and since each value that can be substituted for x will produce only one value for the expression. Recall from Chapter 3 that the domain of a function f(x) is the set of all real numbers that can be substituted for x. Since a rational expression is undefined when its denominator equals zero, we define the domain of a rational function as follows. The domain of a rational function consists of all real numbers except the value(s) of the variable that make(s) the denominator equal zero. Therefore, to determine the domain of a rational function we set the denominator equal to zero and solve for the variable. Any value that makes the denominator equal to zero is not in the domain of the function.

Examples For f(x) = x2 25 x 2 find f(4). find x so that f(x) = 0. determine the domain of the function. Determine the domain of each rational function. 2x 3 f(x) = x 2 8x + 12 g(a) = a + 4 10

All of the operations that can be performed with fractions can also be done with rational expressions. We begin our study of these operations with rational expressions by learning how to write a rational expression in lowest terms. Fundamental Property of Rational Expressions If P, Q, and C are polynomials such that Q 0 and C 0 then P C QC = P Q Writing a Rational Expression in Lowest Terms Completely factor the numerator and denominator. Divide the numerator and denominator by the greatest common factor.

Examples Write each rational expression in lowest terms. 6t 48 t 2 8t b 2 5b 2 6b 8 x 3 + 27 4x 4 12x 3 + 36x 2

Do you think that x 4 is in lowest terms? Let s look at it more closely to 4 x understand the answer. Therefore, x 4 4 x = 1. We can generalize this result as x 4 4 x = x 4 1( 4 + x) = 1 (x 4) 1 (x 4) = 1 a b b a = 1 The terms in the numerator and denominator differ only in sign. They divide out to 1.

Examples Write each rational expression in lowest terms. 15b 5a 2a 6b 12 3y 2 y 2 10y + 16

We multiply rational expressions the same way we multiply rational numbers. Multiply numerators, multiply denominators, and simplify. Multiplying Rational Expressions If P Q and R T are rational expressions, then P Q R T = P R QT To multiply two rational expressions, multiply their numerators, multiply their denominators, and simplify. Multiplying Rational Expressions Factor. Reduce and multiply. All products must be written in lowest terms.

Examples Multiply. x 3 14y 7y2 6 x 8 r 2 25 r 2 9r r2 + r 90 10 2r

To divide rational expressions we multiply the first rational expression by the reciprocal of the second rational expression. Dividing Rational Expressions If P Q and R are rational expressions with Q, R, and T not equal to zero, T then P Q R T = P Q T R = P T QR Multiply the first rational expression by the reciprocal of the second rational expression.

Examples Divide. r 6 35t 9 r4 14t 3 x 2 8x 48 7x 2 42x x 2 16 x 2 10x + 24 5a 2 + 34a + 24 a 2 (5a + 4) 2

Section 7.2: Adding and Subtracting Rational Expressions Recall that to add or subtract fractions, they must have a common denominator. Similarly, rational expressions must have common denominators in order to be added or subtracted. In this section, we will discuss how to find the least common denominator (LCD) of rational expressions. Finding the Least Common Denominator (LCD) Factor the denominators. The LCD will contain each unique factor the greatest number of times it appears in any single factorization. The LCD is the product of the factors identified in Step 2.

Examples Find the LCD of each group of rational expressions. 5 k, 8k k + 2 12 p 2 7p, 6 p 7 x x 2 25, 8 x 2 + 10x + 25 9 r 8, 5 8 r

Writing Rational Expressions as Equivalent Expressions with the Least Common Denominator Identify and write down the LCD. Look at each rational expression (with its denominator in factored form) and compare its denominator with the LCD. Ask yourself, What factors are missing? Multiply the numerator and denominator by the missing factors to obtain an equivalent rational expression with the desired LCD. Multiply the terms in the numerator, but leave the denominator as the product of factors.

Examples Identify the least common denominator of each pair of rational expressions, and rewrite each as an equivalent expression with the LCD as its denominator. 9 7r, 4 5 21r 2 6 y + 4, 8 3y 2 d 1 d 2 + 2d, 3 d 2 + 12d + 20

We can generalize the procedure for adding and subtracting rational expressions that have a common denominator as follows. Adding and Subtracting Rational Expressions If P Q and R Q are rational expressions with Q 0 then P Q + R Q = P + R Q and P Q R Q = P R Q

Examples Add or subtract, as indicated. k 2 + 2k + 5 (k + 4)(k 1) + 5k + 7 (k + 4)(k 1) 20x 9 5x 14 4x(3x + 1) 4x(3x + 1)

If we are asked to add or subtract rational expressions with different denominators, we must begin by rewriting each expression with the least common denominator. Then, add or subtract. Simplify the result. Steps for Adding and Subtracting Rational Expressions with Different Denominators Factor the denominators. Write down the LCD. Rewrite each rational expression as an equivalent rational expression with the LCD. Add or subtract the numerators and keep the common denominator in factored form. After combining like terms in the numerator ask yourself, Can I factor it? If so, factor. Reduce the rational expression, if possible.

Examples Add or subtract, as indicated. 7 12t 3 + 4 9t 6 r 5 + r2 17r r 2 25 1 x 2 36 x + 4 6 x 3k k 2 + 13k + 40 2k 3 k 2 + 7k 8

Section 7.3: Simplifying Complex Fractions In algebra we sometimes encounter fractions that contain fractions in their numerators, denominators, or both. Such fractions are called complex fractions. Some examples of complex fractions are 5 8 3 4 1 2 + 1 3 3 5 4 2 x 2 y 1 y 1 x 7k 28 3 k 4 k Complex Fraction A complex fraction is a rational expression that contains one or more fractions in its numerator, its denominator, or both. A complex fraction is not considered to be an expression in simplest form. In this section, we will learn how to simplify complex fractions to lowest terms.

Examples Simplify. 7k 28 3 k 4 k 6x x 2 81 x 2 2x + 18

Simplify a Complex Fraction with One Term in the Numerator and Denominator To simplify a complex fraction containing one term in the numerator and one term in the denominator: Rewrite the complex fraction as a division problem. Perform the division by multiplying the first fraction by the reciprocal of the second. (We are multiplying the numerator of the complex fraction by the reciprocal of the denominator.)

When a complex fraction has more than one term in the numerator and/or the denominator, we can use one of two methods to simplify. Simplify a Complex Fraction Using Method 1 Combine the terms in the numerator and combine the terms in the denominator so that each contains only one fraction. Rewrite as a division problem. Perform the division by multiplying the first fraction by the reciprocal of the second.

Examples Simplify using method 1. 5 a + 3 ab 1 ab + 2 8 k 1 k + 5 3 k + 5 + 5 k

Simplify a Complex Fraction Using Method 2 Identify and write down the LCD of all of the fractions in the complex fraction. Multiply the numerator and denominator of the complex fraction by the LCD. Simplify.

Examples Simplify using method 2. 5 a + 3 ab 1 ab + 2 8 k 1 k + 5 3 k + 5 + 5 k

Example If a rational expression contains a negative exponent, rewrite it with positive exponents and simplify. Simplify. 4a 2 + b 1 a 3 + b x 2 3y 1 1 + x 1

Section 7.4: Solving Rational Equations A rational equation is an equation that contains a rational expression. Expressions vs. Equations The sum/difference of rational expressions does not contain an = sign. To add or subtract, rewrite each expression with the LCD, and keep the denominator while performing the operations. An equation contains an = sign. To solve an equation containing rational expressions, multiply the equation by the LCD of all fractions to eliminate the denominators, then solve.

How to Solve a Rational Equation If possible, factor all denominators. Write down the LCD of all of the expressions. Multiply both sides of the equation by the LCD to eliminate the denominators. Solve the equation. Check the solution(s) in the original equation. If a proposed solution makes a denominator equal 0, then it is rejected as a solution. It is very important to check the proposed solution. Sometimes, what appears to be a solution actually is not.

Examples Solve each equation. x 2 + 1 = 24 x 7 t + 1 + 2t t + 1 = 3 3y y 4 1 = 12 y 4

Examples Solve each equation. x x + 6 3 x + 2 = 6 3x x 2 + 8x + 12 11 6h 2 + 48h + 90 = h 3h + 15 + 1 2h + 6

A proportion is a statement that two ratios are equal. We can solve a proportion as we have solved the other equations in this section, by multiplying both sides of the equation by the LCD. Or, we can solve a proportion by setting the cross products equal to each other.

Examples Solve each proportion. 9 y = 5 y 2 5 x 2 25 = 4 x 2 + 5x

Examples In Section 2.3, we learned how to solve an equation for a specific variable. Next we discuss how to solve for a specific variable in a rational expression. Solve y = a for c. bc + d Solve 1 a + 1 b = 1 c for a.

Putting It All Together Simplify: 2x 2 7x + 5 1 x 2 Subtract: a a + 3 2 a Multiply: a 3 8 9 3a + 6 a 2 4

Solve: r r + 3 + 5 = 12 r + 3 Simplify: y 2 25 8y 3y + 15 6y For f(x) = 3x 4 x 2 + 5x find x so that f(x) = 0. determine the domain of the function.