Rational Algebraic Expressions and Equations

Rational Algebraic Expressions

• Rational algebraic expressions are written in fraction form, represented as ( \frac{n}{d} ), where ( n ) and ( d ) are polynomials.
• The denominator of a rational expression must not equal zero to be defined.

Rational Algebraic Equations

• Rational algebraic equations comprise one or more rational expressions in their structure.
• An algebraic expression can be expressed as a quotient of two polynomials, making it a rational algebraic equation.

Simplification of Expressions

• Simplifying ( \frac{16p^2}{28p} ) results in ( \frac{4p}{7} ).
• Simplifying ( \frac{15n}{30n^3} ) yields ( \frac{1}{2n^2} ).
• Simplification of ( \frac{x + 6}{x^2 + 5x - 6} ) simplifies to ( \frac{1}{x - 1} ).
• Simplifying ( \frac{2r - 4}{r - 2} ) leads to the result of 2.

Finding the Least Common Denominator (LCD)

• The LCD of ( \frac{2}{x} + \frac{1}{2} ) is ( 2x ).
• The LCD of ( 3x ) and ( 2x ) is ( 6x ).
• The LCD of ( 2x ) and ( 2x ) is ( 4x ).

Polynomials in Rational Expressions

• Both the numerator and denominator in rational algebraic equations consist of polynomials.

Translating Word Expressions to Algebraic Forms

• The sum of a number and 10 is represented as ( x + 10 ).
• Six more than a number is expressed as ( x + 6 ).
• The equation for 17 added to a number yielding 3 is ( x + 17 = 3 ).
• The quotient of a number and 9 is represented as ( \frac{x}{9} ).
• One-half of a number leading to 14 is expressed as ( \frac{x}{14} ).
• Twenty-five times a number is shown as ( 25x ).
• The ratio of a number to 16 is ( \frac{x}{16} ).
• An expression stating that 3 less a number amounts to 10 is ( 3 - x = 10 ).
• Twenty-six added to a number is expressed as ( x + 26 ).
• Seven less a number is represented as ( 7 - x ).