4-1 Polynomial Functions
DEFINITION of POLYNOMIAL in one VARIABLE: A polynomial in one variable,x, is an expression of the form
DEGREE:
EXAMPLE 1
Determine if each expression is a polynomial in one variable. If the expression is a polynomial in one variable, state the degree.
a) c4 + 2a2+ 4c
b) w4 - w2 + 5w5 - 6w3
c) 3x2 + 3/x + 8
If P(x) is a polynomial then P(x) = 0 is a polynomial equation.
A _________ is a solution to the equation.
The ________ is a value of x for which P(x) = 0.
EXAMPLE 2
Determine if 3 is a root of x3 - 2x2 - 5x + 6 = 0
The root may be an imaginary root. i = _____
____________ + ______________ = ____________________
Complex Numbers have the form ___________
If b = 0, then the number is a ____________
If a = 0 and b € 0, then the complex number is a ___________________
FUNDAMENTAL THEOREM OF ALGEBRA:
The degree of a polynomial indicates how many roots are ___________
COROLLARY TO THE FTA:
The graph of a polynomial function with _________degree _______ cross the x-axis.
The graph of a function with _________ degree _____________________ cross the x-axis.
EXAMPLE 3
State the number of complex roots of the equation x3 + 2x2 - 8x = 0. Then find the roots and graph the related polynomial function.
EXAMPLE 4
Write the polynomial equation of least degree with roots -3,2i,and -2i.
Assignment Day 1 181(5-33)
4-2 Quadratic Equations and Inequalities
There are several ways to solve quadratic equations:
1) COMPLETING THE SQUARE:
Trinomial Square: a2 +2ab + b2 = (a + b)2
In order to complete the square, the coefficient of a2 must be 1.
EXAMPLE 1
a) z2 + 4z = 96 b) 3x2 - 11x - 4 = 0
2) QUADRATIC FORMULA
EXAMPLE 2
Solve 4x2 - 8x + 3 = 0 by using the quadratic equation and graph the function.
DICRIMINANT :
Discriminant Nature of the Roots
EXAMPLE 3
Determine the discriminant of x2 - 6x + 13 = 0. Use the quadratic formula to find the roots. Then graph the function.
CONJUGATES:
COMPLEX CONJUGATES THM:
EXAMPLE 4
Graph y > x2 + 8x - 20
Assignment Day 1) 192(1-14)
2) 192(15-32)
4-3 Remainder and Factor Theorem
REMAINDER THEOREM:
EXAMPLE 1
Let P(x) = x3 + 3x2 - 2x - 8. Show that the value of P(-2) is the remainder when P(x) is divided by x + 2.
SYNTHETIC DIVISION
Redo example 1 using synthetic division.
EXAMPLE 2
Use synthetic division to divide m5 - 3m2 - 20 by m - 2.
FACTOR THEOREM
EXAMPLE 3
Let P(x) = x3 - 4x2 - 7x + 10. Determine if x - 5 is a factor of P(x).
Depressed polynomial:
Assignment Day 1) 199(4-30)
4-4 The Rational Root Theorem
RATIONAL ROOT THEOREM
INTEGRAL ROOT THEOREM
EXAMPLE 1
Find the roots of x3 + 6x2 + 10x + 3 = 0
EXAMPLE 2
Find the roots of x3 + 6x2 - 13x - 6 = 0
DESCARTES' RULE OF SIGNS:
EXAMPLE 3
State the number of possible complex zeros, the number of positive real zeros, and the number of possible negative real zeros for h(x) = x4 - 2x3 + 7x2+ 4x - 15.
EXAMPLE 4
Find the zeros of M(x) = x4 + 4x3 + 3x2 - 4x - 4. Then graph the function.
Assignment Day 1) 205(1-2,5-10)
2) 206(11-24)
4-5 Locating the Zeros of a Function
LOCATION PRINCIPLE
EXAMPLE 1
Determine between which consecutive integers the real zeros of f(x) = x3 + 2x2 - 3x - 5 are located.
EXAMPLE 2
Approximate to the nearest tenth the real zeros of f(x) = x4 - 3x3 - 2x2 + 3x - 5. Then sketch the graph of the function, given that the relative maximum is at (0.4,-4.3), and the relative minima are at (-0.7,-6.8) abd (2.5,-17.8).
UPPER BOUND THEOREM
EXAMPLE 3
Find a lower bound of the zeros of f(x) = x4 - 3x3 - 2x2 + 3x - 5.
Assignment Day 1) 213(1-8)
2) 213(9-26)
4-6 Rational Equations and Partial Fractions
EXAMPLE 1
EXAMPLE 2
PARTIAL FRACTIONS
EXAMPLE 3
Decompose ______________ into partial fractions
EXAMPLE 4
EXAMPLE 5
Assignment Day 1) 220(5-12)
2) 221(13-32)
4-7 Radical Equations and Inequalities
EXAMPLE 1
Solve 5 + /x - 4 = 2
EXAMPLE 2
Solve 3 = 3/ x + 4 + 12
EXAMPLE 3
Solve / 3x + 4 - / 2x - 7 = 3
EXAMPLE 4
Solve / 5x + 4 ¾ 8
Assignment Day 1) 228(5-25)