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

Dividing Rational Expressions:

Slope of a Line:

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

Graphing Lines Using the Intercept Method: