Transcript Chapter 4 – Polynomials and Rational Functions

Chapter 4 – Polynomials and
Rational Functions
4.1 Polynomial Functions
Def: A polynomial in one variable, x, is an expression of the form
a0 xn  a1xn1  ...  an2 x2  an1x  an  0 . The coefficients a0, a1, a2, …, an represent
complex numbers (real or imaginary), a0 is not zero, and n represents a
nonnegative integer.
Def: The degree of a polynomial in one variable is the greatest exponent of its
variable.
Def: If a function f is a polynomial in one variable, then f is a polynomial
function.
Def: If p(x) represents a polynomial, then p(x) = 0 is called a polynomial
equation.
Def: A root of the equation is a value of x for which the value of the polynomial
p(x) is 0. It is also called a zero.
Ex: Determine if each expression is a polynomial in one variable. If so, state the
degree.
a. w4  w2  5w5  6w3
4
2
b. c  2a  4c
Ex: Determine whether 3 is a root of x3  2 x2  5x  6  0
c. 3x 2 
3
8
x
What is an imaginary number?
What is a complex number?
Fundamental Theorem of Algebra
Every polynomial equation with degree greater than zero has at least one root in
the set of complex numbers
Corollary to the Fundamental Theorem of Algebra
Every polynomial p(x) of degree n can be written as the product of a constant k and
n linear factors.
p( x)  k  x  r1  x  r2  x  r3  ... x  rn 
Thus, a polynomial equation of degree n has exactly n complex roots, namely r1, r2,
r3, …, rn.
Relationship with degree and roots:
Ex: State the number of complex roots of the equation x3 + 2x2 – 8x = 0. Then
find the roots and graph the related polynomial function.
Ex: Write the polynomial equation of least degree with roots -3 and 2i.
4.2 Quadratic Equations and Inequalities
Ex: Solve each equation by completing the square.
a. x 2  3x  88  0
b. 0  10 x  50 x  1500
2
Quadratic Formula
The roots of a quadratic equation of the form ax2 + bx + c = 0 with a not equal to
zero are given by the following formula.
b  b 2  4ac
x
2a
Ex: Solve 4x2 – 8x + 3 = 0 using the quadratic formula. Then graph
the related function.
b2 – 4ac > 0
b2 – 4ac = 0
b2 – 4ac < 0
Discriminant
two distinct real roots
exactly one real root (double root)
no real roots (imaginary roots)
Ex: Determine the discriminant of x2 – 6x + 13 = 0. Use the
quadratic formula to find the roots. Then graph the related function.
Ex: Graph y > x2 + 8x - 20
4.3 The Remainder and Factor Theorems
The Remainder Theorem
If a polynomial p(x) is divided by x – r, the remainder is a constant, p(r), and
p ( x)  ( x  r )  q ( x )  p (r )
where q(x) is a polynomial with degree one less than the degree of p(x).
Example: 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.
Ex: Use synthetic division to divide m5 – 3m2 – 20 by m – 2.
The Factor Theorem
The binomial x – r is a factor of the polynomial p(x) if and only if p(r) = 0.
Ex: Let p(x) = x3 – 4x2 – 7x + 10. Determine if x – 5 is a factor of p(x).
4.4 The Rational Root Theorem
Rational Root Theorem:
Let a0 xn  a1xn1  ...  an2 x2  an1x  an  0 represent a polynomial equation of degree n
with integral coefficients. If a rational number p/q, where p and q have no
common factors, is a root of the equation, then p is a factor of an and q is a
factor of a0.
Example:
0  2r  r  25
3
Possible values for p:
Possible values for q:
Possible rational roots:
2
Integral Root Theorem:
n
n1
2
Leta0 x  a1x  ...  an2 x  an1x  an  0 represent a polynomial equation that has
leading coefficients of 1, integral coefficients, and an  0 . Any rational roots of
this equation must be integral factors of an.
Ex: Find the roots of x3 + 6x2 +10x +3 = 0.
Descartes’ Rule of Signs
Suppose p(x) is a polynomial whose terms are arranged in descending powers of
the variable. Then the number of positive real zeros of p(x) is the same as the
number of changes in sign of the coefficients of the terms, or is less than this by
an even number. The number of negative real zeros of p(x) is the same as the
number of changes in sign of the coefficients of the terms of p(-x), or is less than
this by an even number.
Ex: 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.
Ex: Find the zeros of M(x) = x4 +4x3 +3x2 – 4x – 4. Then graph the function.
4.5 Locating the Zeros of a Function
The Location Principle:
Suppose y = f(x) represents a polynomial function. If a and b are two numbers
with f(a) negative and f(b) positive, the function has at least one real zero
between a and b.
Ex: Determine between which consecutive integers the real zeros of f(x) = x3 +
2x2 – 3x -5 are located.
Ex: 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) and (2.5, -17.8).
Upper Bound Theorem
Suppose c is a positive integer and p(x) is divided by x – c. If the resulting
quotient and remainder have no change in sign, then p(x) has no real zeros
greater than c. Thus, c is an upper bound of zeros of p(x).
Ex: Find a lower bound of the zeros of f(x) = x4 – 3x3 – 2x2 +3x – 5.
4.6 Rational Equations and Partial Fractions
t4
3
16
Ex: Solve

 2
t
t  4 t  4t
Ex: Solve
h h

1
10 20
Decompose
Solve
x  11
x2  2 x  3
 x  3 x  4 
2
 x  5  x  6 
0
into partial fractions
Solve:
1
5 1


4 a 8a 2
4.7 Radical Equations and Inequalities
Solve:
5 x 4  2
Solve:
3  3 x  4  12
Ex: Solve
3x  4  2x  7  3
Ex: Solve
5x  4  8