Theorems on polynomial functions

Remainder theorem, Factor theorem, Rational Zeroes theorem

1. Theorems on Polynomial Functions
PSHS Main Campus
July 13, 2012
PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 1 / 7

2. Remainder Theorem
Remainder Theorem
When a polynomialP(x)is divided by(xa), the remainder isP(a).
Examples
1
Findf(3)iff(x) = 2x
3
5x
2
8x+ 17.
2
Findg(5)ifg(x) =x
4
22x
2
+ 13x+ 15.
PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 2 / 7

3. Factor Theorem
Factor Theorem
A polynomial functionf(x)has a factor(xa)if and only iff(a) = 0.
Examples
1
Conrm that(x5)is a factor ofx
4
3x
3
+ 7x
2
60x125.
2
Show that2x
4
11x
3
+ 14x
2
+ 9x18is divisible byx
2
5x+ 6.
PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 3 / 7

4. Rational Zero Theorem
Rational Root Theorem
RZT/RRT
If:
1
f(x)is a polynomial function with integral coecients,
2p
q
, a rational number in simplest terms is a zero off(x), i.e.,
f

p
q

= 0,
then:
1
pis a factor of the constant term
2
qis a factor of the leading coecient
Example
Find the rational zeros off(x) = 12x
3
8x
2
3x+ 2.
PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 4 / 7

5. Corollary of RZT
RZT foran= 1
If the leading coecient of a polynomial function with integral coecients
is1, then any rational zeros off(x)are integers.
Example
Find the rational zeros off(x) =x
4
+ 3x
3
+ 2x
2
3x3.
PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 5 / 7

6. Fundamental Theorem of Algebra
Fundamental Theorem of Algebra
Every polynomial function with complex coecients has at least one zero
in the set of complex numbers.
Implication of the FTA
Every polynomial function with nhas ncomplex zeros.
Example
Find ALL zeros off(x) =x
4
x
3
x
2
x2.
PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 6 / 7

7. Homework 12
1
Use the remainder theorem to evaluate the functions below:
1f(4=5),f(x) = 5x
3
9x
2
+ 3x11
2g(1=3),g(x) = 6x
3
3x
2
+ 5x8
2
Use the factor theorem to determine if the rst expression is a factor
of the second expression.
1x1;2x
3
x
2
+ 2x3
24x1;x
3

9
4
x
2
+
23
2
x
11
4
3
Find the values ofaandbsuch thatx
3
2ax
2
+bx3is divisible by
x
2
x2.
4
Findaandbsuch thatax
3
bx
2
+ 45x+ 54 = 0has3as a root and
yields a remainder of 12 when divided byx+ 1.
PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 7 / 7