Download 3x3+5x2+8x+7 by 3x+2

October 23, 2009
2.4 Real Zeros of Polynomial Functions
Divide using long division.
3x3+5x2+8x+7
by
3x+2
October 23, 2009
Find the remainder when f(x)=x3 - 3x + 4 is divided by x + 2.
October 23, 2009
Remainder Theorem:
If a polynomial f(x) is divided by x-k, then the remainder
is r = f(k).
Find the remainder when f(x)=3x2+ 7x - 20 is divided by x + 4.
October 23, 2009
Q: What does it mean to get a remainder of 0?
Factor Theorem:
A polynomial function f(x) has a factor of x - k iff f(k)=0.
Is x - 3 a factor of x3- x2- x- 15?
October 23, 2009
Synthetic Division
3
2
Divide 2x - 3x - 5x - 12 by x - 3 using synthetic division.
October 23, 2009
Rational Zeros Theorem:
If there are rational zeros in the polynomial:
n
n-1
f(x) = anx + an-1x
+ ... + a0
then they will be in the set
p
q
where p = factors of the constant, a0
and
q = factors of the leading term coefficient, an
October 23, 2009
Find the rational zeros of
f(x) = 3x3 + 4x2 - 5x - 2
October 23, 2009
Upper and Lower Bound Tests for Real Zeros
Suppose f(x) is divided by x - k using synthetic division.
1. If k≥ 0 and every number in the last line is nonnegative (pos. or
zero), then k is an upper bound for the real zeros of f.
2. If k ≤ 0 and the numbers in the last line alternate nonnegative
and nonpositive, then k is a lower bound for the real zeros of f.
October 23, 2009
Prove that k = 5 is an upper bound for the real
zeros of f(x) = 2x3 - 5x2 - 5x - 1
Prove that k = -4 is a lower bound for the real
zeros of f(x) = 3x3 - x2 - 5x - 3