     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 9: 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:  Examples : # Solving Inequalities: 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:  #     Example:  Simplify Example: Simplify # Operations with Polynomials:

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

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

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:  # Dividing Rational Expressions:      Slope of a Line:    Slope y-intercept Form of the Equation of a Line:   Graphing Lines Using the Intercept Method:  