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 polynomial P (x) is divided by (x − a), the remainder is P (a).
Examples
1 Find f (3) if f (x) = 2x3 − 5x2 − 8x + 17.
2 Find g(−5) if g(x) = x4 − 22x2 + 13x + 15.
PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 2/7

3. Factor Theorem
Factor Theorem
A polynomial function f (x) has a factor (x − a) if and only if f (a) = 0.
Examples
1 Confirm that (x − 5) is a factor of x4 − 3x3 + 7x2 − 60x − 125.
2 Show that 2x4 − 11x3 + 14x2 + 9x − 18 is divisible by x2 − 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 coefficients,
p
q, a rational number in simplest terms is a zero of f (x), i.e.,
2
p
f q = 0,
then:
1 p is a factor of the constant term
2 q is a factor of the leading coefficient
Example
Find the rational zeros of f (x) = 12x3 − 8x2 − 3x + 2.
PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 4/7

5. Corollary of RZT
RZT for an = 1
If the leading coefficient of a polynomial function with integral coefficients
is 1, then any rational zeros of f (x) are integers.
Example
Find the rational zeros of f (x) = x4 + 3x3 + 2x2 − 3x − 3.
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 coefficients has at least one zero
in the set of complex numbers.
Implication of the FTA
Every polynomial function with degree n has exactly n complex zeros.
Example
Find ALL zeros of f (x) = x4 − x3 − x2 − x − 2.
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:
1 f (−4/5), f (x) = 5x3 − 9x2 + 3x − 11
2 g(1/3), g(x) = 6x3 − 3x2 + 5x − 8
2 Use the factor theorem to determine if the first expression is a factor
of the second expression.
1 x − 1 ; 2x3 − x2 + 2x − 3
2 4x − 1 ; x3 − 4 x2 + 23 x −
9
2
11
4
3 Find the values of a and b such that x3 − 2ax2 + bx − 3 is divisible by
x2 − x − 2.
4 Find a and b such that ax3 − bx2 + 45x + 54 = 0 has 3 as a root and
yields a remainder of 12 when divided by x + 1.
PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 7/7