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4.5 Quadratic Equations
Zero of the Function- a value where f(x) = 0 and the graph of the function intersects the x-axis
Zero Product Property- for all numbers a and b, if ab = 0, then a = 0, b = 0, or both a = 0 and b = 0
4.5 Quadratic Equations
4.5 Quadratic Equations
4.6 Completing the Square
-By Completing the Square:
1. Set up ax2 + bx = c (divide by a if needed) y a if
2. Add (b/2)2 to both sides
3. Factor the left side
4.7 Quadratic Formula
-
4.8 Complex Numbers-Complex Number — any number that can be written in form a + bi;
4.8 Complex Numbers
Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
Multiplication:(a + bi)(c + di) = ac + adi + bci + bdi2 = (ac - bd) + (ad + bc)i
Multiplying Conjugates: (a + bi)(a - bi) = a2 + b2
Division:
5.1 Polynomial Functions
Monomial- a real number, a variable, or a product of a real number and one or more variables with whole number exponents
Degree of a Monomial- in one variable is the exponent of the variable
Polynomial- monomial or a sum of monomials
Degree of a Polynomial- in one variable is the greatest degree among the its monomial terms
5.1 Polynomial Functions
-Standard Form of a Polynomial Function:
1. Coefficients (a) must be real #’s2. Exponents must be positive integers3. Domain = All Real #’s4. Degree of a polynomial function is the highest
degree of x (n)
f x a x a x a x a x ann
nn( ) ...
11
22
1 0
5.1 Polynomial Functions
1. Graphs of polynomials are smooth & continuous ; a turning point is where the graph changes directions
2. Leading Term Test for End Behavior:
a) if n is odd and an > 0 if n is odd and an < 0
b) if n is even and an > 0 if n is even and an < 0
3. The graph can have at most n – 1 turning points
lim ( ) ; lim ( )
lim ( ) ; lim ( )x x
x x
f x f x
f x f x
lim ( ) ; lim ( )
lim ( ) ; lim ( )x x
x x
f x f x
f x f x
5.2 Polynomials, Linear Factors, and Zeros
-Real Zeros of Polynomial Functions:
x = a is a zero of function f means x = a is a solution of the equation f(x) = 0 means
(x – a) is a factor of f(x) means (a,0) is an x-intercept of the graph of f
-A function f can have at most n real zeros
-Multiplicity of a zero—the # of times (x – a) occurs as a factor of f(x)
“Even Multiplicity” Graph touches the x-axis“Odd Multiplicity” Graph crosses the x-axis
5.2 Polynomials, Linear Factors, and Zeros
-always measured on the x-axis-always named from Left to Right-always open brackets ( ) -Functions ONLY
Local and Absolute Extrema:-local (relative) Maximum —the value of f(x) at the turning
point when a graph goes from increasing to decreasing-local (relative) Minimum—the value of f(x) at the turning
point when a graph goes from decreasing to increasing
5.3 Solving Polynomial Equations
Factored Polynomial- a polynomial is factored when it is expressed as a the product of monomials and polynomials
Factoring by Grouping- when the terms and factors of a polynomial are grouped separately so that the remaining polynomial factors of each group are the same
5.3 Solving Polynomial Equations
Factoring by Grouping-
Sum or Difference of Cubes-
5.4 Dividing Polynomials
-Synthetic Division:Given: ax3 + bx2 + cx + d divided by x – k
Synthetic division method:
1.Add columns2.Multiply by k
a b c dk
a
ka
remainder
5.4 Dividing Polynomials
-Remainder Theorem:
If a polynomial f(x) is divided by (x – k) then the remainder is r = f(k)
-Factor Theorem:
1. If f(c) = 0, then (x – c) is a factor of f(x)2. If (x – c) is a factor of f(x), then f(c) = 0
5.5 Theorems About Roots of Polynomial Equations
-Rational Zero Theorem:f x a x a x a x a x an
nn
n( ) ...
11
22
1 0
Given: integer coefficients and a 0 and 0
Every rational zero of f(x) has the form p/q , where:
1. p and q have no common factors other than 12. p is a factor of the constant term (a0) 3. q is a factor of the leading coefficient (an)
Complex Conjugate Theorem: if (a + bi) is a zero of f(x), then (a – bi) is also a zero
Fundamental Theorem of Algebra- A polynomial of degree n has exactly n [real and non-real (complex)] zeros (roots). Some zeros may be repeated.
-A polynomial of degree n has exactly n linear factors of the form f(x) = a(x – c)(x – d)(x – e)…(x – n)
-A polynomial of degree n has at least one complex zero
x = a is a zero of function f means x = a is a solution of the equation f(x) = 0 means
(x – a) is a factor of f(x)
5.6 The Fundamental Theorem of Algebra