Download Section 2-4

Section 2-4
Real Zeros of Polynomial
Functions
Section 2-4
• long division and the division algorithm
• the remainder and factor theorems
• reviewing the fundamental connection for
polynomial functions
• synthetic division
• rational zeros theorem
• upper and lower bounds
Long Division
• long division for polynomials is just like
long division for numbers
• it involves a dividend divided by a divisor
to obtain a quotient and a remainder
• the dividend is the numerator of a fraction
and the divisor is the denominator
Division Algorithm
• if f(x) is the dividend, d(x) is the divisor,
q(x) the quotient, and r(x) the remainder,
then the division algorithm can be stated
two ways
f ( x)  d ( x)  q( x)  r ( x)
or
f ( x)
r ( x)
 q( x) 
d ( x)
d ( x)
Remainder Theorem
• if a polynomial f (x) is divided by x – k, then
the remainder is f (k)
• in other words, the remainder of the division
problem would be the same value as plugging
in k into the f (x)
• we can find the remainders without having to
do long division
• later, we will find f (k) values without having
to plug k into the function using a shortcut for
long division
Factor Theorem
• the useful aspect of the remainder theorem is
what happens when the remainder is 0
• since the remainder is 0, f (k) = 0 which
means that k is a zero of the polynomial
• it also means that x – k is a factor of the
polynomial
• if we could find out what values yield
remainders of 0 then we can find factors of
polynomials of higher degree
Fundamental Connection
For a real number k and a polynomial function
f (x), the following statements are equivalent
•
•
•
•
k is a solution (or root) of the equation f (x) = 0
k is a zero of the function f (x)
k is an x-intercept of the graph of f (x)
x – k is a factor of f (x)
Synthetic Division
• finding zeros and factors of polynomials
would be simple if we had some easy way
to find out which values would produce a
remainder of 0 (long division takes too
long)
• synthetic division is just that shortcut
• it allows us to quickly divide a function
f (x) by a divisor x – k to see if it yields a
remainder of 0
Synthetic Division
• it follows the same steps as long division
without having to write out the variables
and other notation
• it is really fast and easy
• if a zero is found, the resulting quotient is
also a factor, and it is called the depressed
equation because it will be one degree
less than the original function
Synthetic Division
Divide 2 x  3x  5 x  12 by x  3
3
2
Synthetic Division
Divide 2 x  3x  5 x  12 by x  3
3
2
3
2
-3
-5
- 12
Synthetic Division
Divide 2 x  3x  5 x  12 by x  3
3
2
2
3
2
-3
-5
- 12
6
9
12
3
4
0
Synthetic Division
Divide 2 x  3x  5 x  12 by x  3
3
2
2
3
2
-3
-5
- 12
6
9
12
3
4
0
The remainder is 0 so x – 3 is a factor and
the quotient, 2x2 + 3x + 4, is also a factor
Rational Zeros Theorems
• if you want to find zeros, you need to
have an idea about which values to test in
S.D. (synthetic division)
• the rational zeros theorem provides a list
of possible rational zeros to test in S.D.
• they will be a value  p
where,
q
p must be a factor of the constant
q must be a factor of the leading term
Finding Possible Rational Zeros
Find all the possible rational zeros of
f ( x)  3 x  4 x  5 x  2
3
2
p  2 and q  3
factors of p: 1, 2
factors of q: 1, 3
1 2
possible rational zeros:  1, 2, ,
3 3
Upper and Lower Bounds
• a number k is an upper bound if there are
no zeros greater than k; if k is plugged into
S.D., the bottom line will have no sign
changes
• a number k is a lower bound if there are
no zeros less than k; if k is plugged into
S.D., the bottom line will have alternating
signs (0 can be considered + or -)
Upper and Lower Bounds
• if you are looking for zeros and you come
across a lower bound, do not try any
numbers less than that number
• if you are trying to find zeros and you
come across an upper bound, do not try
any numbers greater than that number