Rational Expressions Simplifying

Simplifying Rational Expressions The objective is to be able to simplify a rational expression.

Polynomial – The sum or difference of monomials. Rational expression – A fraction whose numerator and denominator are polynomials. Domain of a rational expression – the set of all real numbers except those for which the denominator is zero. Reduced form – a rational expression in which the numerator and denominator have no factors in common.

Divide out the common factors Factor the numerator and denominator and then divide the common factors

Dividing Out Common Factors Step 1 – Identify any factors which are common to both the numerator and the denominator.  The numerator and denominator have a common factor.  The common factor is the five.

Dividing Out Common Factors Step 2 – Divide out the common factors.  The fives can be divided since 5/5 = 1  The x remains in the numerator.  The (x-7) remains in the denominator

Factoring the Numerator and Denominator  Factor the numerator.  Factor the denominator.  Divide out the common factors.  Write in simplified form.

Factoring Step 1: Look for common factors to both terms in the numerator.  3 is a factor of both 3 and 9.  X is a factor of both x 2 and x. Step 2: Factor the numerator xx x () 

Factoring Step 3: Look for common factors to the terms in the denominator and factor.  The denominator only has one term. The 12 and x 3 can be factored.  The 12 can be factored into 3 x 4.  The x 3 can be written as x x xx xx ()  

Divide and Simplify Step 4: Divide out the common factors. In this case, the common factors divide to become 1. Step 5: Write in simplified form. x x  3 4 2

You Try It Simplify the following rational expressions.

Problem 1 Divide out the common factors. Write in simplified form. 3 8 x z

Problem 2 Factor the numerator and denominator Divide out the common factors. Write in simplified form. You Try It

Problem 3 Factor the numerator and denominator Divide out the common factors. Write in simplified form. You Try It

Problem 4 You Try It Factor the numerator and denominator Divide out the common factors. Write in simplified form.

Problem 5 You Try It Factor the numerator and denominator Divide out the common factors. Write in simplified form.

Restrictions on Rational Expressions For what value of x is undefined? It is undefined for any value of “x” which makes the denominator zero. The restriction is that x cannot equal 5.

YOU TRY IT What are the excluded values of the variables for the following rational expressions?

Problem 1 Solution y  0 z  0

Problem 2 Solution 2x - 12 = 0 2x = x = 12 2x  2 = 12  2 x = 6 ANSWER x  6

Problem 3 Solution C 2 + 2C - 8 = 0 (C-2)(C+4) = 0 C-2 = 0 or C + 4 = 0 C-2+2 = C = C = 2 or C = -4 Answer C  2 C  -4

Assignment Problems