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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 an1 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 an1 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)