Rational Expressions and Functions (12) (ELCA)

Properties of Rational Expressions and Functions

Rational Expressions

• Rational Expressions – is the quotient or ratio of two polynomials with the denominator not equal to zero.

Example – $\frac{2}{3}$, $3a + 5$, $\frac{x - 3}{2x^2 - 2}$, $\frac{y + 2}{5y}$, and $\frac{x - 2}{x + 1}$.

Domain and Range

• Range – is a rational set of all real numbers that can be used to find the value of zero. (y) (Dependent Variable) (The Zeros of an equation)
• Domain – is a rational set of all real numbers that can be used in place of the variable. (x) (Independent Variable)

How to Write Sets
Set Builder Notation	{x|x ≠ -5}	x equals all reals except -5
Interval Notation	(-∞, -5) U (-5, ∞)

Reducing Rational Expressions

• $\frac{2x^2 - 18}{x^2+x-6} = \frac{2(x^2 - 9}{(x - 2)(x + 3)} = \frac{2(x+3)(x-3)}{(x - 2)(x + 3)} = \frac{2(x-3)}{x-2}$
• $\frac{2a^3 - 16}{16 - 4a^2} = \frac{2a^3 - 16}{-4a^2+16} = \frac{2(a^3-8}{-4(a^2-4)} = \frac{2 (a-2)(a^2+2a+4)}{-4(a+2)(a-2)} = \frac{2(a^2+2a+4)}{-4(a+2)}$
• Rule of Thirds – $a^3-8 = a^3 - 2^3 = (a-2)(a^2+2a+4)$

Strategy for Reducing Rational Expressions

1) All reducing is done by dividing out common factors.

2) Factor the numerator and denominator completely to see the common factors.

3) Use the quotient rule to reduce a ratio of two monomials involving exponents.

4) We may have to factor out a common factor with a negative sign to get identical factors in the numerator and denominator.

Factoring

Difference of Squares

$a^2-b^2=(a-b)(a+b)$

Examples:

1. $y^2-36=y^2-6^2=(y-6)(y+6)$
2. $9x^2-25=3^2 \times x^2 - 5^2= (3x-5)(3x+5)$

Difference of Cubes

$a^3-b^3=(a-b)(a^2+ab+b^2)$
Examples:

1. $x^3-8=x^3-2^3=(x-2)(x^2+2x+4)$
2. $8x^3-64=2^3 \times x^3 - 4^3 = (2x-4)(4x^2+8x+16)$

Sum of Two Cubes

$a^3+b^3=(a+b)(a^2-ab+b^2)$

Polynomials

Examples:

• $2x^2 + 9x + 4 =$

$2x^2 + x + 8x + 4=$
$(2x+1) x + (2x+1) 4 = (2x+1) (x+4)$

• $2x^2 + 5x - 12 =$

$2x^2 - 3x + 8x - 12 =$
$(2x-3) x + (2x-3) 4 = (2x-3)(x+4)$

• $6w^2 - w - 15 =$

$6w^2 - 10w + 9w - 15 =$
$2w (3w-5) + 3 (3w-5) = (2w+3)(3w-5)$

Building Up Denominator

Fractions without identical denominators can be converted to equivalent fractions with a common denominator by reversing the procedure for reducing fractions to the lowest term.

Examples:

• $\frac{2}{7} = \frac{\text{?}}{42}$

$\frac{2}{7} \times \frac{6}{6} = \frac{12}{42}$

• $\frac{5}{3a^2b} = \frac{\text{?}}{9a^3b^4}$

$\frac{5}{3a^2b} \times \frac{3ab^3}{3ab^3} = \frac{15ab^3}{9a^3b^4}$

Rational Functions

A rational expression can be used to determine the value of a variable.

Examples:

• Find R(3) for $R(x) = \frac{3x-1}{x^2 - 4}$

$R(3) = \frac{3 \times (3) -1}{(3)^2 - 4} = \frac{8}{5}$

Real Life Application: Average Cost

A car maker spent $700 million to develop a new SUV, which will sell for $40,000. If the cost of manufacturing the SUV is $30,000 each, then what rational function gives the average cost to developing and manufacturing x vehicles? Compare the average cost per vehicle for manufacturing levels of 10,000 vehicles and 100,000 vehicles.

The polynomial $30,000x + 700,000,000$ gives the cost in dollars for development and manufacture of x vehicles. The average cost per vehicles is:

$AC(x) = \frac{30,000x + 700,000,000}{x}$

$AC(10,000) = \frac{30,000 \times (10,000) + 700,000,000}{(10,000)} = 100,000$

$AC(100,000) = \frac{30,000 \times (100,000) + 700,000,000}{(100,000)} = 37,000$

Multiplication and Division

Multiplying Rational Expressions

One multiplies rational expressions by multiplying their numerators and denominators.

Example:

$\frac{6}{7} \times \frac{14}{15} = \frac{84}{105} \div \frac{21}{21} =\frac{4}{5}$

It is more common to reduce rational expressions before multiplying.

Example:

$\frac{6}{7} \times \frac{14}{15} = \frac{2 \times 3 \times 2 \times 7}{7 \times 3 \times 5} = \frac{2 \times 2}{5} = \frac{4}{5}$

If $\frac{a}{b}$ and $\frac{c}{d}$ are rational numbers, then $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

More Examples:

$\frac{3a^8b^3}{6b} \times \frac{10a}{a^2b^6} = \frac{3a^8b^3}{2 \times 3b} \times \frac{2 \times 5a}{a^2b^6} = \frac{a^8b^3}{b} \times \frac{5a}{a^2b^6} = \frac{5a^9b^3}{a^2b^7} = \frac{5a^7}{b^4}$

Dividing Rational Expressions

When dividing rational numbers, you must multiply by the reciprocal or multiplicative inverse of the divisor.

Example:

$\frac{3}{4} \div \frac{15}{2} = \frac{3}{4} \times \frac{2}{15} = \frac{3}{2 \times 2} \times \frac{2 \times 1}{3 \times 5} = \frac{1}{2} \times \frac{1}{5} = \frac{1}{10}$

$\text{If } \frac{a}{b} \text{ and } \frac{c}{d} \text{ are rational numbers with } \frac{c}{d} \not = 0 \text{ , } \text{ then: } \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Example:

$\frac{25 - x^2}{x^2 + x} \div \frac{x - 5}{x^2 - 1} = \frac{25 - x^2}{x^2 + x} \times \frac{x - 5}{x^2 - 1} =$

$\frac{-1(x - 5)(5+x)}{x(x+1)} \times \frac{(x + 1)(x - 1)}{x - 5} = \frac{-1(5+x)}{x} \times \frac{(x - 1)}{1} =$

$\frac{-1(5x +1)(x - 1)}{x} = \frac{-x^2 - 4x + 5}{x}$

Addition and Subtraction

In order to add or subtract addition of subtraction problems, one must find a common denominator.

$\text{If b } \not = 0 \text{ then } \frac{a}{b} + \frac{c}{b} = \frac{a + c}{b} \text{ and } \frac{a}{b} - \frac{c}{b} = \frac{a-c}{b}$

Examples:

$\frac{2x - 3}{8} - \frac{x - 2}{6} = \frac{3(2x - 3)}{24} - \frac{4(x - 2)}{24} = \frac{(6x - 9) - (4x - 8)}{24} = \frac{6x - 9 - 4x + 8}{24} = \frac{2x - 1}{24}$

$\frac{5x^2}{30xy} - \frac{30x}{80y} = \frac{x}{6y} - \frac{3x}{8y} = \frac{4x}{24y} - \frac{9x}{24y} = \frac{-5x}{24y}$

Complex Fractions

Complex Fractions – a fraction that has rational expressions in the numerator, the denominational, or both.

Examples:

$\frac{\frac{1}{2} + \frac{1}{3}}{\frac{1}{4} + \frac{1}{5}} = \frac{\frac{5}{6}}{\frac{9}{20}} = \frac{5}{6} \div \frac{9}{20} = \frac{5}{6} \times \frac{20}{9} = \frac{5 \times 2 \times 10}{2 \times 3 \times 9} = \frac{5 \times 10}{3 \times 9} = \frac{50}{27}$

$\frac{3 - \frac{2}{x}}{\frac{1}{x^2}-\frac{1}{4}} = \frac{3 - \frac{2}{x}}{\frac{1}{x^2}-\frac{1}{4}} \times \frac{4x^2}{4x^2} = \frac{12x^2 - 8x}{4 - x^2} = \frac{12x^2 - 8x}{(2 - x)(2 + x)}$

$\frac{3a^\text{-1} - 2^\text{-1}}{1 - b^\text{-1}} = \frac{3a^\text{-1} - 2^\text{-1}}{1 - b^\text{-1}} \times \frac{2ab}{2ab} = \frac{6b - ab}{2ab - 2a} = \frac{6b - ab}{2a(b - 1)}$