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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
𝑥2
+ 𝑏𝑥 +
1. Set up
2. Add
(b/2)2
2
𝑏 2
= (𝑥 + )
2
4𝑥 2 − 20𝑥 + 17 = 0
-By Completing the Square:
ax2
𝑏
2
+ bx = c (divide by a if needed)
to both sides
𝑥2
5
2
− 5𝑥 +
3. Factor the left side ( 𝑥 − )2 = 2
5 2
2
→
𝑥2
=
− 5𝑥 =
−17
y
4
−17
5 2
+( )
4
2
5
2
𝑥− = ± 2
a if
4.7 Quadratic Formula
-
4.8 Complex Numbers
-Complex Number — any number that can be written
in form a + bi; 𝑎 + 𝑏𝑖 = 𝑎2 + 𝑏 2
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
(a + bi)(a - bi) = a2 + b2
Multiplying Conjugates:
Division:
bc − ad
𝑖
𝑐 2 +𝑑 2
a + b𝑖
c + d𝑖
=
a + b𝑖
c + d𝑖
∙
c − d𝑖
c− d𝑖
=
ac + bd
+
𝑐 2+ 𝑑2
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:
f ( x)  an x  an1 x
n
n 1
...a2 x  a1 x  a0
2
1. Coefficients (a) must be real #’s
2. Exponents must be positive integers
3. Domain = All Real #’s
4. Degree of a polynomial function is the highest
degree of x (n)
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 
lim f ( x )  ; lim f ( x )  
x 
x 
lim f ( x )  ; lim f ( x )  
x 
x 
lim f ( x )  ; lim f ( x )  
x 
x 
lim f ( x )  ; lim f ( x )  
x 
x 
3. The graph can have at most n – 1 turning points
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 columns
2.Multiply by k
k
a
b
c
d
ka
a
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)  an x  an1 x
n
n 1
...a2 x  a1 x  a0
2
Given: integer coefficients and a 𝑎𝑛 ≠ 0 and 𝑎0 ≠ 0
Every rational zero of f(x) has the form p/q , where:
1. p and q have no common factors other than 1
2. 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
5.6 The Fundamental Theorem of
Algebra
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 ≥ 1 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)