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Rational Root Theorem
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Period
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How do we find the factors of a polynomial?
1.
Factor the following quadratics
(a)
p( x)  x 2  x  30
(b)
p ( x)  4 x 2  5 x  6
2.
How did you decide which numbers to try as factors? Did you use the Factor Theorem? How would you extend your
thinking to factor higher order polynomials?
3.
Given the polynomial
f ( x)  x 4  3x3  15x 2  19 x  30 . If we were to factor this polynomial
as ( x  a)( x  b)( x  c)( x  d ) , what would be the possible values of a, b, c and d? Try to find the values of a, b, c,
and d by using the Factor Theorem and dividing out each factor as you successfully find it. Graph the function on your
calculator to check your answers.
4.
When the zeros are all integers it is easy to use the calculator, but if they are not all integers it is hard to tell if it factors or
if the solutions are irrational, (i.e. one of the factors is a quadratic that can’t be reduced further using only integer
coefficients)
The polynomial equation 3x  5 x  26 x  8  0 has a solution which is not an integer, but it is a rational number. To
use your calculator to help you find the solutions in such a case, set the xscl to 1/3. (Why would we do this?) Set the
window to:
Xmin = -5 Ymin = -20
Xmax = 5
Ymax = 60
Xscl = 1/3 Yscl = 1
3
2
If we look at the graph we see that it crosses the x-axis one notch to the right at x 
twelve notches to the left at
x  4 .
1
, six notches to the right at x  2 ,
3
Write this equation in factored form and prove the factoring works.
5.
Use a similar technique to solve 2 x  x  31x  15  0 (what is your x-step here?)
3
2
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6.
If x

p
is a zero of a polynomial then (qx-p) is a factor of the polynomial. Why is this?
q
Fill in the blanks:
The Rational Roots Theorem:
Given a polynomial f(x) the only possible rational solutions of the equation f(x) = 0 are
x
p
. Where p is a factor
q
of…………………….…… and q is a factor of ……………………………
7.
Use the rational roots theorem and the factor theorem to factor the following polynomials (you may use your calculator as
much as you like).
(a) x4-3x2-4x+12
(b) 2x4-5x3-14x2+5x+12
The Remainder Theorem.
If a polynomial p(x) is divided by a linear binomial ( x  c) the remainder will always be p(c).
8.
(a)
9.
Perform the following divisions and then evaluate p(c) in each case to satisfy yourself of the truth of this theorem:
p( x)  x3  x 2  x  5 divided by ( x  1)
134
What would the remainder be if 2 x
(b) p(x)=3x3-12x2-9x+1 divided by (x-5)
 7 x 45  x9  4 is divided by ( x  1) ?
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