September 22, 2014
2.3 Real Zeros of Polynomial Functions
Objectives: 1.Use long division and synthetic division to divide polynomials, and
2.
find rational and real zeros of polynomial functions
n
lle
a
Ch
ge
Use long division to find 7213 - 61
quotient
divisor
61
7213
dividend
*Remember to subtract!
Note: If no remainder, then the divisor is a factor of the
dividend. That means you can use division to find factors and slant
asymptotes (Sect. 2.7).
Let's try it on polynomials!
Divide 2x3- 5x2+x-8 by x-2.
s
ent
oti
u
q
al
rti
*pa
x-2 2x3- 5x2+x-8
Note: If no remainder, then the divisor is a factor of the dividend.
That means you can use division to find factors and slant asymptotes.
(Sect. 2.7)
September 22, 2014
Division Algorithm
*Use to check your answer.
For all polynomials f(x) and d(x) such that the degree of
d(x) < degree of f(x) and d(x) = 0, there exist unique polynomials
q(x) and r(x) such that
f(x) = d(x)q(x) + r(x), where r(x) = 0
or the degree of r<degree of d.
dividend
divisor
quotient
remainder
3
Ex: Divide x - 2 by x-1
Another way to look at it...
Pull
f(x) q(x) r(x)
+
=
d(x)
d(x)
Synthetic Division
*Note: Divisor must be x-c
Let's do 1st example again using synthetic division!
Ex: Divide 2x3- 5x2+x-8 by x-2.
1
-8
coefficients
of dividends
mu
ltipl
y
2 -5
bring down
2
add
remainder
coefficients of quotient
September 22, 2014
Now, evaluate f(x) = 2x3- 5x2+x-8 when x = 2.
What do you notice?
ol!
o
C
The Remainder Theorem
If a polynomial f(x) is divided by x-k, the remainder is r = f(k).
*Proof in Appendix C
Ex: Use the Remainder Thm to find g(2) when g(x)=x3- 2x2- 4x+1
September 22, 2014
The Factor Theorem
A polynomial f(x) has a factor (x-k) iff f(k)= 0.
*Use to test whether x-k is a factor of the polynomial by evaluating f(k).
Ex: Show that x-1 is a factor of x4-1.
Ex:Show that x-2 and x+3 are factors of
f(x) = 2x4+7x3-4x2-27x-18
then find remaining factors of f(x).
Pull
Complete fac
f(x)=(x-2)(x+