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Rational Expressions 1

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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

                                                                                                                                                            {2, 3}  adds to 5

               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

            Add to “-5”

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

            Add to “-9”

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:

 

Text Box: Simplifying Rational Expressions:
·	Factor the numerator and denominator if possible
·	Reduce to lowest terms by dividing out common factors
·	State any restrictions – remember the denominator cannot equal 0.
 

 

 

 

 

 

 

 

 


Example 1:

                                                                                        

Solution:

 

           

 

 

                       

 

 

 

 

 

 

           

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