     Home Real Numbers Radicals Inequalities & Absolute Value Polynomials Rational Expressions 1 Rat'l. Exp. 2 - mult&div Rat'l. Exp. 3 - add&subt Slope & Equations of Lines Review&Test ### UNIT 1  : PRECALCULUS PREP

LESSON 5:  RATIONAL EXPRESSIONS 1

# Rational Expressions:

Rational expressions are quotients whose numerator and denominator are both polynomials [monomials, binomial, trinomial, etc…]

Examples:   Review of Basic Factoring methods:

1. Common Factoring:

Factor 6x3 – 15x

Solution:  6x3 – 15x = 3x(2x2 – 5)                                           ** Find the HIGHEST COMMON FACTOR for each term ---  “3x”

** Divide “3x” into each term to get the second factor  --- “2x2 – 5”

** Check by expanding

2. Difference of Squares: Formula --  a2 – b2 = (a – b)(a + b)

Factor 49x2 – 64y2

Solution:  49x2 – 64y2 = (7x – 8y)(7x + 8y)

Factor x2 – 9y2

Solution: x2 – 9y2 = (x – 3y)(x + 3y)

3. Simple Trinomials: Form  x2 + bx + c  [Coefficient of x2 is 1]

a) Factor x2 + 5x + 6

Solution: Recall  x2 + 5x + 6 = (x + __ )(x + __ )                     ** We need two numbers that multiply to +6 and add to +5

** Check all pairs of factors of 6:          {1, 6}  adds to 7

Hence  x2 + 5x + 6 = (x + 3 )(x + 2)                          ** Check by expanding

b) Factor x2 – 2x – 35

Solution: Recall  x2 - 2x - 35 = (x + __ )(x + __ )                    ** We need two numbers that multiply to -35 and add to -2

** Check all pairs of factors of 35:        {1, 35}  no combination adds to “-2”

{5, 7}  -7 + 5 = -2

Hence  x2 – 2x - 35 = (x + 5 )(x – 7)                         ** Check by expanding

4. Hard Trinomials: Form  ax2 + bx + c  [Coefficient of x2  does not equal 1]

a) Factor 6m2 – 5m – 4

Solution:

We use the method of decomposition (although there are other methods)

We decompose the middle term “-5m” into two parts using the two clues:

Multiply to (6)(-4) = -24   and

Strategy: List all pairs of factors of “-24” and find the pair that adds to “-5”     ** {1, 24}  cannot obtain “-5” with these two factors; 1+24=25; 1-24= -23

**  {2, 12} cannot obtain “-5” with these two factors

**  {3, 8}  -8 + 3 = -5 Choose these two factors

Hence  6m2 – 5m – 4 = 6m2 – 8m + 3m –4                                                       ** “-5m” broken into two parts “-8m + 3m”

= 2m(3m – 4) + 1(3m – 4)                                           ** Group by twos and common factor

= (3m – 4)(2m + 1)                                                      ** check by expanding

b) Factor 2x2 – 9x + 4

Solution:

We decompose the middle term “-9x” into two parts using the two clues:

Multiply to (2)(4) = +8   and

Strategy: List all pairs of factors of “+8” and find the pair that adds to “-9”       ** {2, 4}  cannot obtain “-9” with these two factors; 2+4=6; 2-4= -2

** {1, 8}  -8 + -1 = -9  Choose these two factors

Hence  2x2 – 9x + 4 = 2x2 –8x – 1x + 4                                                           ** “-9x” broken into two parts “-8x – 1x”

= 2x(x – 4) – 1(x – 4)                                                   ** Group by twos and common factor

= (x – 4)(2x – 1)                                                          ** check by expanding

ALTERNATE METHOD - FACTORING TRINOMIALS USING THE X-METHOD:

Example 1: Example 2: Example 3: Example 4: Simplifying Rational Expressions: Example 1:    Solution:     