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Rational Expressions and Functions (12) (ELCA)

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Properties of Rational Expressions and FunctionsEdit

Rational ExpressionsEdit

  • 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 RangeEdit

  • 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 ExpressionsEdit

  • \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 Thirdsa^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.

FactoringEdit

Difference of SquaresEdit

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 CubesEdit

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 CubesEdit

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

PolynomialsEdit

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 DenominatorEdit

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 FunctionsEdit

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 CostEdit

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 DivisionEdit

Multiplying Rational ExpressionsEdit

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 ExpressionsEdit

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 SubtractionEdit

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 FractionsEdit

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)}

Division of PolynomialsEdit

Dividing by a MonomialEdit

Dividing by a BinomialEdit

Synthetic DivisionEdit

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