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