College Algebra Notes, Section P.6

Notes, Lesson P.6
Rational Expressions

When a monomial is divided by a monomial, cancellation and exponent laws allow us to attempt to simplify the result.

Example:

	Given Problem.
	Because both numerator and denominator are monomials, think of the problem as being in 3 separate parts.
	Simplify each part.

When a polynomial is divided by a monomial, you must divide each term of the polynomial by the monomial. Usually this is done by decomposing the fraction into its addends.

Example:

	Given Problem.
	Begin by "backing up", that is, writing the problem as 3 addends with a common denominator.
	Simplify each part, and look for like terms.

When a polynomial is divided by a polynomial, one method to use is the long division of polynomials.

Example:

	Given Problem. It is very important that when you do a long division problem of this type that you write the dividend in descending order. And if a degree is missing in the sequence, you need to leave space for this missing term. Usually this is done by writing the variable with the appropriate exponent and a zero coefficient.
	In your division, you must think division of monomials. So here you should be thinking, "What times 3x is 6x cubed?" - Answer: 2x squared. So now multiply (3x+2) times 2x squared, subtract, and bring down the next term.
	In your division, you must think division of monomials. "What times 3x is -9x squared?" - Answer: -3x. So now multiply (3x+2) times -3x, subtract and bring down next term.
	"What times 3x is -6x?" - Answer: -2. Now multiply (3x+2) times -2 and subtract. There is no remainder.
	Multiply the divisor by the quotient and see if it checks with the dividend. It does.

Rational polynomials are polynomials in fraction form or in the process of being divided. It is imperative that you are adept at combining these rational polynomials in various ways. We will be combining rational (fractional) polynomials using addition, subtraction, multiplication, and division.

Addition of rational polynomials: The far left column is provided as a comparison, showing that manipulating rational polynomials follows the same rules and procedures that are followed using rational arithmetic.

Rational Numbers	Rational Algebraic Expressions	Explanation
		Given Problem.
		Find least common multiple of the two denominators. This is the LCD or Least Common Denominator.
		Find new numerators for the new common denominator.
		Simplfy numerators.
		Combine (add) numerators over common denominator.

Subtraction:

Rational Numbers	Rational Algebraic Expressions	Explanation
		Given Problem.
		Find least common multiple of the two denominators. This is the LCD or Least Common Denominator.
		Find new numerators for the new common denominator.
		Combine (add) numerators over common denominator.

Multiplication:

Rational Numbers	Rational Algebraic Expressions	Explanation
		Given Problem.
		Factor numerators and denominators
		Cancel any like factors
		Simplify if possible

Division:

Rational Numbers	Rational Algebraic Expressions	Explanation
		Given Problem.
		Rewrite as multiplication problem with the reciprocal of the second fraction.
		Factor numerators and denominators
		Cancel any like factors
		Neither of these examples can be simplified further.

Some Notes from Class