Download 2. f(x) = 2x 4+7x3-4x2-27x-18 a) Is (x-5) a likely factor

November 18, 2013
Precalc Warm Up # 11-1
2
1. Divide x3 + 5x + 2x - 8 by x + 2. Was it a factor?
2.
f(x) = 2x4+7x3-4x2-27x-18
a) Is (x-5) a likely factor of f(x)?
b) Is (3x+2) a likely factor of f(x)?
c) Is (x-2) a likely factor of f(x)?
d) Factor f(x) completely
November 18, 2013
2
1. Divide x3 + 5x + 2x - 8 by x + 2. Was it a factor?
November 18, 2013
2.
f(x) = 2x4+7x3-4x2-27x-18
a)
b)
c)
d)
Is (x-5) a likely factor of f(x)?
Is (3x+2) a likely factor of f(x)?
Is (x-2) a likely factor of f(x)?
Factor f(x) completely
November 18, 2013
Questions on #10-5?
Divide using long division.
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Divide using synthetic division
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Use synthetic division to show that x is a solution of the third
degree polynomial equation, and factor completely.
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DESCARTES'S RULE OF SIGNS:
1. the number of POSITIVE REAL ZEROs is either equal to
the number of variations in sign of f(x) or less than that by an
even integer.
2. the number of NEGATIVE REAL ZEROS is either equal to
the number of variation in signs of f(-x) or less than that by
an even integer.
f(x) = 2x4+7x3-4x2-27x-18 had four zeros: 2,-3, -1, and -3/2
f(x) has 1 change in sign, which indicates that it has 1 positive
real zero.
f(-x) = 2(-x)4 + 7(-x)3 -4(-x)2-27(-x) - 18
= 2x4
-
7x3
- 4x2 + 27x - 18
therefore f(-x) has 3 changes in sign, indicating that it has 3
or 1 negative real zeros.
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HOW CAN THIS HELP US SOLVE
HIGHER DEGREE EQUATIONS?
Use Descartes's rule of signs to find all solutions to
x4-3x3+x2+3x-2 = 0
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We have also been taking advantage of the rational
zero test
Rational Zero Test:
If f(x) = anxn + an-1xn-1 + ...+a1x + a0 has integer
coefficients, then every rational zero of f is p/q,
where p is a factor of constant term a0 and q is a
factor of the leading coefficient an
That looks more complicated than it is! It is just
observing that if f(x) = 2x3+ 4x -5, then (3x+7) isn't
a good factor guess. Which means -7/3 isn't going to
be a zero.
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List all of the possible rational zeros of f. Using
those and Descartes's rule of signs, find all of zeros
and graph
f(x)= -3x3+20x2-36x+16
November 18, 2013
List all of the possible rational zeros of f. Using
those and Descartes's rule of signs, find all of zeros
and graph
f(x)= -3x3+20x2-36x+16
November 18, 2013
One last trick to have up your sleeve......
LOWER AND UPPER BOUND RULE
Let f(x) be a polynomial with real coefficients and a positive
leading coefficient. Suppose f(x) is divided by (x-c) using
synthetic division.
1. If c>0 and each number in the last row is either
positive or zero, then c is an UPPER BOUND for the zeros
2. If c<0 and the numbers in the last row are alternately
positive and negative (0 counts either way), then c is a
LOWER BOUND for the zeros
November 18, 2013
Let's use this on finding the real zeros of
f(x) = 3x3-2x2+15x-10
3 variations of signs, so either 3 or 1 positive real
zeros
f(-x) = -3x3-2x2-15x-10
No variations of signs, so no negative real zeros
Try 1. If it is an upper bound, what does this tell us?
November 18, 2013
#11-1 p. 211 box, 53
The test is this Friday, and it will cover 3-1 --->3-6