Download 3.3 Factoring Polynomials

 3.3 Factoring Polynomials I.
Factor out the Greatest Common Factor a. b. c. Ex 1. 12 x 3  18 x 4 Ex. 2 3a(2a  5b)  7b(2a  5b) II. Factoring by Grouping a. b. c. Ex.1 2 x 3  3x 2  14 x  21 Ex.2 18 x 2 y  3xy 2  18 xy  3 y 2 3.4 Factoring Binomials I.
Difference of Two Squares a.
Prime polynomial Ex.1 75 x 3  3 x Ex.2 y 4  16 II. Sum and Difference of Two Cubes Ex.1 Ex.2 3.5 Factoring Trinomials I.
Trinomials of the form ax 2  bx  c , where a  1 Ex1. x2  7x  6 Ex.2 x 2  x  20 Ex.3 2 x 3  12 x 2  16 x II. Trinomials of the form ax 2  bx  c , where a  1 Ex.1 2 x 2  5x  3 Ex.2 12 x 2  13x  4 Ex.3 16 x 2  34 x  15 3.6 Factoring Summary 1. Examine the terms to determine if there is a common factor. If so, factor out the greatest common factor. 2. Consider the number of terms: a. Two terms Difference of squares? Difference of cubes? Sum of cubes? b. Three terms Leading coefficient of 1? Leading coefficient not equal to 1? c. Four or more terms Factor by grouping 3. Examine all the factors to determine if any factor can be factored further. I.
3.7 Solving Quadratic Equations by Factoring Definition Any equation that can be written in the standard form ax 2  bx  c  0 , where a, b, and care real numbers with a  0 , is called a quadratic equation in the variable x. Zero Product Theorem: Let a and b be real numbers. If a  b  0 , then a = 0 or b = 0. II.
Steps to Solving Quadratic Equation a. b. c. d. e. Ex.1 256  16t 2  400 Ex.2 (2 x  1)( x  3)  9 3.8 Applications 4.1 Integer Exponents I Laws of Integer Exponents a. Product Law: b. Power to a Power Law: c. Product to a Power Law: d. Quotient Law of Exponents: e.
Quotient to a Power Law of Exponents: f.
Zero Exponent Rule: 4.2 Reducing Rational Expressions to Lowest Terms I.
Definition: Any expression that can be written as the quotient of two polynomials, where the denominator is not equal to zero, is called a rational expression. II.
Reduction Property of Fractions 5x 8 y 2 z 6
Ex.1 5 4 9 4x y z
Ex.2 2 x 2  3 x  35
x 2  4x  5
Ex.3 x 2  4 x  21
9  x2
4.3 Multiplying and Dividing Rational Expressions I. Multiplication Property of Fractions II. Division Property of Fractions Exs. 4.4 Adding and Subtracting Rational Expressions 1. Like denominators a. Combine Expressions (Add/Subtract) b. Simplify/Reduce i.
2 6
 11 11
ii.
x
3
 2
x  x  12 x  x  12
2
2. Unlike denominators a. Factor denominators b. Find the Least Common Denominator (LCD) c. Combine Expressions (Add/Subtract) d. Simplify/Reduce i.
1
5

18 24
ii.
x 1
x2
 2
6x  7 x  2 2x  7 x  3
2
4.5 Complex Fractions I. Definition Any fraction that contains a fraction in its numerator and/or denominator is called a complex fraction II. Steps (Simplify) – First Method A. B. Exs. 1. 2. C 3. III. Steps (Simplify) Second Method A. B. Exs. 1. 2. 3. 4.8 Equations Involving Rational Expressions (7‐Step Method) 1. Rational Equations (equations involving rational expressions and operations) a. 7‐Step Method i. Reduce (Factor) expressions ii. Multiply both sides of equation by Least Common Denominator (LCD) iii. Eliminate Grouping Symbols (Parentheses) iv. Isolate the Unknown v. Factor equation vi. Solve equation vii. Check for extraneous solutions b. Ex. Ex) 3x  1
3
2x  5
 2
 2
x  4 x  3 x  x  12 x  3 x  4
2
1) 2) 3) 4) 5) 6) 7) Remember My Education Is For Skillful Calculations Reduce Multiply Eliminate Isolate Factor Solve Check 4.9 Applications Ex. 2 Ex.3 Ex.6 5.1 Rational Exponents I.
Review of Exponent Rules II. Definition Ex.2 Ex.4 Let a > 0 and n be a positive even integer. The principle nth root of a, denoted n
 1
a , is the positive number such that  a n   a  
1
n
III. Definition Let a be a real number and n be a positive odd integer. The nth root of a, n
 1n 
denoted a , is the real number such that  a   a .  
 
1
n
Ex.5 Ex.6 IV. Definition Let m and n be positive integers with m
n
m
 1
  a n   
a
a n  am
Or m
m
being reduced to lowest terms; then n
 
Provided a n is a real number. 1
1
n
Ex.7 V. Definition Ex.9 Ex.10 Let m and n be positive integers with Then a

m
n

1
 mn 
a 
 
 
m
being reduced to lowest terms and a  0 ; n
5.2 Radicals I.
Let n be a positive integer greater than 1 and a be a real number; then 1
an  n a a. b. c. d. Ex1. Ex2. II. If n is a positive even integer and a is a real number, then n
an  a If n is a positive odd integer and a is a real number, then Ex.3 Ex.4 Ex.5 n an  a III.Let m and n be positive integers and a be a real number; then m
m
a n  n a m or a n 
 a
n
m
Provided that n a is a real number. 1. 2. Ex.6 Ex.7 Ex.8 5.3 Simplifying Radical Expressions I.
Multiplication Property of Radicals Let n be a natural number; then n ab 
negative. Exs. II. Simplifying a Radical 1. 2. 3. n
a  n b provided that both a and b are not bot h Exs. III.Division Property of Radicals Let n be a natural number; then n
Exs. a

b
n
a
n
b
provided that both a and b are not negative and b  0 . IV. Rationalizing the denominator 1. 2. 3. Exs. V. Simplifying a Radical 1. 2. 3. 4. 5.4 Operations with Radical Expressions I.
Def Like Radicals Exs. 5.5 More Operations with Radical Expressions I.
Conjugates Exs. 5.6 Radical Equations I.
If a = b, then a n  b n , where n is any natural number. Extraneous solutions II. Steps to Solve Radical Equation 1. 2. 3. Exs. 4. 5.7 Complex Numbers I. Def i   1 and i 2  1 If r > 0, then  r 

r  1 r i or i r Exs. II. Def Any number that can be expressed in the form a  bi , where a and b are real numbers and i   1 , is called a complex number. Remarks 1. 2. 3. 4. 5. III.Addition and Subtraction of Complex Numbers Let a, b, c, and d be real numbers and i 
Exs.  1 . IV.Division of Complex Numbers Let a, b, c, and d be real numbers and i 
 1 . Exs. V.Def 1. i0  1 If n is any natural number, i  n 
Exs. 1
in
2.2 Linear Inequalities I.
Def Any inequality that can be written in the form ax  b  0 or ax  b  0 where a and b are real numbers with a  0 , is called a linear inequality in the variable x. II. Def Two inequaities are said to be equivalent if they have the same solution set. III. Additive Property of Inequality Exs. IV.Multiplicative Property of Inequality Exs. V.Interval Notation Remarks 1. 2. 3. Exs. 6.1 The Cartesian Coordinate System I.
Ordered Pair: II. Cartesian Coordinate System/rectangular coordinate system: III. Origin: IV. X‐axis V. Y‐axis VI. Quadrants: VII. Solution to an equation in two variables: 6.2 Relations and Functions I. Relation: II. Domain: III.Range: IV.Def (Function): V.Alternative Definition: VI. Vertical Line Test: 6.3 Function Notation and Combination of Functions I.
Functional Notation: II. Combinations of Functions 6.4 Linear Functions I. Def 1 II. Def II III.Slope Exs. 6.5 Equations of Lines I.
Point‐slope Form: Exs. II. Slope‐intercept Form: Exs. III. Properties of Parallel and Perpendicular Lines 1. 2. Exs. 6.7 Linear Inequalities in Two Variables I.
Def. 1 II. Steps in Graphing Linear Inequalities 1. 2. Exs. 7.3 Solving Quadratic Equations by Completing the Square I.
Extraction of Roots Theorem Exs. II. Steps of Solving Quadratic Equation by Completing the Square 1. 2. 3. 4. 5. Exs. 7.4 Quadratic Formula I.
Exs. Quadratic Formula II. The discriminant If ax 2  bx  c  0 , where a,b, and care rational numbers with a  0 and b 2  4ac is Zero, A positive perfect square, then there is one rational solution, called a double root. then there are two rational solutions. Positive, but not a perfect square then there are two irrational solutions. Negative, then there are two nonreal solutions. If the discriminant is a perfect square, the quadratic is factorable. Exs. 7.5 Quadratic Equation Summary 7.6 Applications I.
Ex.1 II. Ex.2 III. Ex.3 9.3 Applications (Linear Systems of Equations) I.
Ex1. II. Ex2. III. Ex3.