Download Polynomial and rational functions

Theory of Polynomials and Rational Functions
Summary 3
Polynomial Functions: A polynomial function of degree n, where n is a whole
number, is a function of the form
P( x)  a n x n  a n 1x n 1  a n  2 x n  2  . . .  a 2 x 2  a1x  a 0






an is called the leading coefficient
ao is called the constant term.
The y-intercept: of a polynomial function is the constant term.
If n=0 the polynomial function is a constant function,
If n=1 the polynomial function is a linear function
If n=2 the polynomial function is quadratic function.

A power function is a polynomial function of the form y  a n x n where n is a
natural number

The function y  x 2 is a power function and it is called the squaring function. The
graph is a parabola.
Polynomial functions are always continuous and smooth.

Example 1: P( x)  2x 5  3x 4  2x 2  5 is a polynomial function of degree 5, leading
coefficient 2 and constant term 5.
Power functions.


Power functions y  a n x n go through the origin. That is, the x-intercept =0 and the
y-intercept =0.
End-points behavior
Degree n
Odd degree
Odd degree
Even degree
Even degree
Coefficient an
an is positive
an is negative
an is positive
an is negative
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End-points behavior
 ,   and , 
 ,  and ,  
 ,  and , 
 ,   and ,  
Quadratic Functions: Any function of the form y  ax 2  bx  c where a, b, c are real numbers
and a≠0.
By using the technique of completing the square very quadratic function y=ax2+bx+c (Standard form)
2
can be change to the form y  k  a( x  h) where (h, k) gives the vertex of the parabola and x=h is
the line of symmetry.
How to find the vertex of the parabola?
Let y  ax 2  bx  c
Step 1. Subtract c from both members of the equation  y  c  ax 2  bx
b 

Step 2. Factor on the right expression  y  c  a x 2  x 
a 

2
b2
b
1 b
Step 3. Complete the square by adding    
to x 2  x.
2
a
2 a
4a
Step 4. keep the equation balanced .
2
 2 b
b 2 
b2
b 


y c a
a x  x
y
 c  a x  
2

a
4
a
2a 

4a 2
4
a


b2
2
b 2  4ac
b 
b
b 2  4ac

y
 a x    h  
and k 
4a
2a 
2a
4a

Example 2. Find the vertex and the range of the parabola
y  2x 2  7 x  5
Step 1 : y  2x 2  7 x  5  y  5  2x 2  7 x
7 

Step 2 : y  5  2 x 2  x 
2 

2
49
1 7
Step 3 :    
16
 2 2
49
7
49 
49
7


Step 4 : y  5  2 
 2 x 2  x    y  5 
 2 x  
16
2
16 
8
4


2
9
7
7
9
9

7
 y   2 x    h  , k   , vertex is  ,  
8
4
4
8
8

4
-2-
2
Zero or root of P(x): if P(c) =0 then x = c is called a root or zero of P(x).
If c is a real number, then c is an x-intercept of the function. The real zeros of a function
are the values of x where the graph intersects the x-axis.
Division Algorithm: If a polynomial P(x) is divided by a nonzero polynomial d(x), then
there is a quotient q(x) and a remainder polynomial r(x) such that P(x)=d(x)q(x) + r(x).
The Remainder Theorem: If a polynomial P(x) is divided by x-c, then
the remainder is P(c).
The Factor Theorem: the number c is a root or zero of the polynomial
P(x) exactly when x-c is a factor of f(x).
Fundamental Theorem of Algebra: Every non-constant polynomial function
has a root or zero in the complex system (the complex system includes
the real numbers as a subset.)
Factorization Theorem: If P(x) is a polynomial of degree n>0 with
leading coefficient an then P(x) can be factored in n linear factors (not
necessarily distinct) over the complex numbers.
That is, P(x)  an (x  c1(x  c 2 ). . . (x  cn )
Conjugate Roots Theorem: Let P(x) be a polynomial with real coefficients. If the
complex number a + bi is a root then the complex conjugate a - bi is also a root.
Quadratic factor Theorem: If c1 and c2 are zeroes of a polynomial function P(x) then
x 2  (c1  c 2 )x  c1c 2 is a quadratic factor of the polynomial.
Rational roots theorem: If c=
p
is a rational root (in lowest terms) of a polynomial
q
function f(x), then p is a factor or divisor of the constant term a o, and q is a divisor of the
leading coefficient an .
Intermediate value theorem: If P(x) is a polynomial function with real coefficient, and
P(c) and P(d) are opposite in sign, then there exist at least one zero between c and d.
Descartes' rule of signs: let P(x) be a polynomial with real coefficients and a nonzero
constant term, whose terms are arranged in order of decreasing powers of x (with terms
having a zero coefficient deleted).
a) The number of positive real roots of P(x) is either equal to the number of variations of sign of P(x) or is
less than that number by an even integer.
b) The number of negative real roots of P(x) is either equal to the number of variations in sign of P(-x) or is
less than that number by an even integer.
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Multiplicity: If ( x  c)k is a factor of a polynomial function P(x) then c is called a zero
of multiplicity k.
Bounds on the roots of a polynomial: let P(x) be a polynomial with positive leading
coefficient.
a) If c>0 and every number in the last row in the synthetic division of P(x) by x - c
is nonnegative, then c is an upper bound for the roots of P(x), that is, no zero of the function is
greater than c.
b) If c<0 and the numbers in the last row of the synthetic division of P(x) by x - c are alternating
positive and negative, then c is a lower bound for the roots of P(x), that is, no zero is less than
c.
End points behavior of a polynomial function
Degree n is even
Degree n is odd
Leading coefficient an  0
If x   then y  
If x   then y  
If x   then y  
Leading coefficient an  0
If x   then y  
If x   then y  
If x   then y  
How to graph polynomial functions?
1. Determine the end points behavior
2. Find the y-intercept  a 0
3. Use the rational root theorem and synthetic division to find the rational zeroes of P(x).
Approximate any other real zeroes.
3 Express the polynomial in factored form.
4. The real zeroes give the x-intercepts of the graph.
 If c is a real zero of multiplicity n, and n is odd, then the graph crosses the x-axis at
x = c.
 If c is a real zero of multiplicity n, and n is even, then the graph does not cross the
x-axis at x = c, it bounces at that point.
5. Use few other point to complete the graph of y=P(x).
Rational Functions. A rational function is a function of the form R(x) =
P(x )
Q(x )
where P(x) and Q(x) are
polynomial functions.
How to graph rational functions?
Step 1. The zeros of R(x) are the zeros of the numerator P(x) which are not zeros of the denominator
Q(x). The real zeroes of R(x) give the x-intercepts of the function. The rules of even and odd multiplicity
apply as before.
( x  2)( x  3)( x  1) 2
Example 3: R ( x ) 
( x  5)( x  3)
The real zeros or x-intercepts of the function are x=2 and x=-1.
Note that x = -3 is not a zero of the function because x + 3 is a factor of both the numerator and the
denominator.
-4-
Step 2. The zeroes of the denominator are called the Poles of the function.

If Q(c)  0 and P(c)  0 then y=R(x) is undefined at x= c. In this case x = c is a vertical
asymptote of the function
k
k
 If Q(c) =0, and P(c) = 0 then ( x  c) 1 is a factor of the numerator and ( x  c) 2 is a factor of
the denominator. If k1  k 2 , then the function has a hole at x=c. Otherwise the function has a
vertical asymptote at x=c.
( x  2)( x  3)( x  1) 2
Example 4: R ( x ) 
( x  5)( x  3)
The above function has a hole at x=-3 and has a vertical asymptote at x=5
To find the ordinate y, of the hole, cancel the factor (x+3) and compute
The y-intercept (let x=0) is equal to
Example 5: R ( x ) 
y
y
20
7
6
15
( x  2)( x  1)( x  1) 2
( x  5)( x  1) 3
The above function has zeros or x-intercepts at x=2, x=1, and has a vertical asymptotes at x=5 and x=-1
The y-intercept (let x=0) is equal to
y
2
5
Rules of multiplicity for poles:
a) If c is a pole of odd multiplicity and x=c is a vertical asymptote, then the graph of R(x) approaches the
asymptote x=c from opposite sides of the x-axis.
b) If c is a pole of even multiplicity and x=c is a vertical asymptote, then the graph of R(x) approaches the
asymptote x=c from the same side of the x-axis.
Step 3. Find the y-intercept by letting x=0.
Step 4 . If degree of P(x)< degree Q(x) then y=0 (the x-axis) is a horizontal asymptote.
Example : the function R( x ) 
( 2x  3) 2
( x  1)( x  5) 2
as the x  axis as horizontal asymptote
because deg P  deg Q.
Step 5. If degree of P(x)= degree Q(x) then y =
leading coefficient of P(x)
leading coefficient of Q(x )
horizontal asymptote.
-5-
is a
4
( 2x  3) 2
Example : the line y  is a horizontal asymptote of the function R( x) 
3
(3x  1)( x  5)
because deg ree P  deg ree Q.
Step 6. If degree of P(x) >degree of Q(x), then at the ends, the function behaves the same as
the power function
y  ax n , where n  deg ree of P(x)  deg ree of Q( x)
Example : at the ends , the function R( x) 
(3x  2) 3
behaves as the power function
(3x  1))
y  3x 2 because deg ree P  deg ree Q  2
Step 7. If degree of P(x) = degree of Q(x) + 1, then we have a slant asymptote that is obtained
by dividing P(x) by Q(x).
The slant asymptote is y = mx + b, where mx + b is the quotient that results when P(x) is
divided by Q(x).
Find the slant asymptote of the function R(x) 
-2
|
3
-5
4
-6
22
3
-11
26
The slant asymptote is y =2x -11
Step 8. Plot a few selected points.
-2
|
3
-5
4
-6
22
3
-11
26
The slant asymptote is y =2x -11
8. Plot a few selected points.
-6-
3x 2  5x  4
x2