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Real Numbers & Radicals

Basic Algebra & Polynomials

Linear Equations

Inequalities & Absolute Value

Rational Expressions 1

Rat'l. Exp. 2 - mult&div

Rat'l. Exp. 3 - add&subt

Exponents

Slope & Equations of Lines

Review&Test

 

UNIT 1  : ALGEBRA PREP

 LESSON 10: LESSON SUMMARY

 

Review of Basic Exponent Laws:

 

Rule

Example

Explanation

am x an = am+n

32 x 35 = 37

Multiplication Rule - If the bases are the same, add the exponents

Division Rule - If the bases are the same, subtract the exponents

(am)n = amn

(32)3=36

Power Rule – When taking a power of a power, multiply the exponents

(ab)m = amam

(3 x 2)4 = 34 x 24

Power of a Product – Take each factor in the product to that power

Power of a Quotient – Take numerator and denominator to that power

 

Zero Exponents:     Rule:     a0 = 1

 

Examples:  20 = 1;     (-3.4)0 = 1   (x2)0 = 1;            Note that  00 is not defined.

 

 

 

Negative Exponents:    Rule:         or       or   

                                                         

Examples:  ;      ;             

Rational Exponents:

Rule #1:     

 

Examples : 

Rule #2:    

Text Box: Key Points:
·	For fractional exponents, the denominator “n” gives the index of the root
·	The numerator “m” gives the exponent.
·	If n is an odd number, then x can be any real number, positive or negative.
·	If n is even, then x must be positive if we are working in the real number system
·	Eg. -  ,  [not a Real number since n is even]
           

 

 

 

 

 

 

 

 

 

Examples : 

Solving Inequalities:

 

Text Box: KEY IDEAS:
·	Use the same rules as you would in solving equations
·	When multiplying or dividing both sides of the inequality by a negative number, the inequality is reversed.
 

 

 


                                                        

 

 

 Example 1:  Solve      5x –2 < 7x + 8

                                    5x – 7x < 8 + 2

                                          -2x < 10

                                                                   ** Note the inequality reverses when dividing by a negative number

Inequalities Involving Absolute Value:

 

 

Text Box: Absolute Value Inequalities:
 


       

 

 

 

 

 

 

 

 

 

 

 

 

Operations with Radicals:

 

Text Box: 	          
  
                       

                       

 

 

 

 

 

 

 

 

 

 

Example:  Simplify

Example: Simplify

           

 

 

Operations with Polynomials:

 

 

Addition and Subtraction:

 

Example 1: “+” sign preceding brackets – simply drop the brackets and collect like terms

 

a)  (3x2 – 2x + 5) + (5x2 – 3x – 6) = 3x2 – 2x + 5 + 5x2 – 3x – 6                      ** drop brackets

                                                     = 3x2 + 5x2 –2x – 3x +5 – 6                        ** Collect like terms

                                                     = 8x2 – 5x – 1

 

Example 2: “-” sign preceding brackets – multiply each term in the bracket by “-1” and collect like terms

 

(5x2 – 3x + 6) – (2x2 – 7x + 8) = (5x2 – 3x + 6) – 1(2x2 – 7x + 8)                    ** multiply 2nd bracket by “-1”

                                                = 5x2 – 3x + 6 – 2x2 + 7x - 8

                                                = 5x2 – 2x2 – 3x + 7x + 6 – 8                           ** collect like terms

                                                = 3x2 + 4x - 2

Multiplying with Polynomials (Expanding):

 

Example 1: Monomial x Polynomial – multiply each term in bracket by the monomial

 

a) –3x(2x2 – 5x + 7) = -3x(2x2) – 3x(-5x) – 3x(7)                    ** multiply each term by “-3x”

                                = -6x3 + 15x2 – 21x

b)  2x(5x – 3) –5(2x + 7) = 10x2 – 6x – 10x – 35                     ** multiply each term in 1st bracket by “2x” and 2nd bracket by “-5”

                                        = 10x2 – 16x – 35                            ** collect like terms

 

Example 2: Polynomial x Polynomial – multiply each term in 1st  bracket by each term in 2nd  bracket

 

a) (3x + 5)(2x – 7) = 3x(2x – 7) + 5(2x – 7)                                         ** multiply each term in 1st  bracket by each term in 2nd  bracket

                              = 6x2 – 21x + 10x –35                                           ** expand as in previous example

b) (2x + 3)(3x2 – 5x – 2) = 2x(3x2 – 5x – 2) + 3(3x2 – 5x – 2)             ** multiply each term in 1st  bracket by each term in 2nd  bracket

                                       = 6x3 –10x2 – 4x + 9x2 –15x – 6                   ** expand

                                       = 6x3 – x2 –19x – 6                                       ** collect like terms

 

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

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 = -24  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                                                                                                

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.

 

  =                        

Restrictions:   and

 

Multiplying Rational Expressions:

           

Text Box: Multiplying Rational Expressions:
·	Factor the numerators and denominators of all expressions if possible
·	Reduce to lowest terms by dividing out common factors
·	Write answer as a single rational expression
·	State any restrictions – remember the denominators cannot equal 0.
 

 

 

 

 

 

 

 

 


 

Dividing Rational Expressions:

 

 

Text Box: Dividing Rational Expressions:
·	Factor the numerators and denominators of all expressions if possible
·	Multiply by the reciprocal of the second expression
·	Reduce to lowest terms by dividing out common factors
·	Write answer as a single rational expression
·	State any restrictions – remember the denominator cannot equal zero in the first expression and neither the numerator nor the denominator can equal zero in the second expression.
 


           

 

 

 

 

 

 

 

 

 

Adding and Subtracting Rational Expressions:

 

Text Box: Adding and Subtracting Rational Expressions with LIKE DENOMINATORS:
·	Keep the common denominator 
·	Add or subtract the numerators
·	State any restrictions – remember the denominators cannot equal 0.
 

 

 

 

 

 

 

 


 

Text Box: Adding and Subtracting Rational Expressions with UNLIKE DENOMINATORS:
·	Factor each denominator 
·	LCD ---ΰTake the highest exponent of each factor
·	Add or subtract the numerators, simplifying fully
·	State any restrictions – remember the denominators cannot equal 0.
 

 

 

 

 

 

 

 

 

 

 


 

 

Slope of a Line:

                                                                                                    

 

 

                                        

 

             

 

 

Slope y-intercept Form of the Equation of a Line:

 

           

 

 

 

 

 

Graphing Lines Using the Intercept Method:

 

                                             

                                              

 

 

 

 

       

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