2.3 Factoring Polynomials Math Worksheet Page 7
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7Create a flow chart that will describe which strategies you would use to try
13.
C
to factor a polynomial. For each path through the flow chart, give an
example of a polynomial that would follow that path and show its factored
form. Explain how your flow chart could describe how to factor or show
the non-factorability of any polynomial in this chapter.
Extending
4
2
The polynomial
x
2 5x
1 4
is not factorable, but it can be factored by a
14.
form of completing the square:
4
2
x
2 5x
1 4
The first three terms form a
perfect square.
4
2
2
2
5 x
1 4x
1 4 2 4x
2 5x
2
2
2
5 (x
1 2)
2 9x
This is now a difference
2
2
of squares.
1 2 2 3x) (x
1 2 1 3x)
5 (x
2
2
5 (x
2 3x 1 2) (x
1 3x 1 2)
5 (x 2 2) (x 2 1) (x 1 2) (x 1 1)
Use this strategy to factor each polynomial by creating a perfect square.
4
2
4
2
x
1 3x
1 36
x
2 23x
1 49
a)
b)
2
Expanding confirms that
x
2 1 5 (x 2 1) (x 1 1)
and also that
15.
3
2
x
2 1 5 (x 2 1) (x
1 x 1 1).
Make conjectures and determine similar factorings for each expression.
4
n
x
2 1
x
2 1
a)
c)
5
n
n
b)
x
2 1
d)
x
2 y
m
Mersenne numbers are numbers of the form
n 5 2
2 1,
where m is a
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natural number. For example, if
m 5 6,
then
2
2 1 5 64 2 1 5 63,
and if
5
m 5 5,
then
2
2 1 5 32 2 1 5 31.
63 5 3 3 21
is a Mersenne number
that is composite, and
31 5 1 3 31
is a Mersenne number that is prime.
The French mathematician Mersenne was interested in finding the values of
m that produced prime numbers, n.
2
4
2
0
63 5 3(21)
can also be expressed as
(2
2 1) (2
1 2
1 2
),
and
a)
3
3
0
63 5 7(9)
can also be expressed as
(2
2 1) (2
1 2
).
Expand the
expressions that contain powers, treating them like polynomials, to show
6
2
2 1.
that you get
9
3
6
3
0
b) If
m 5 9,
then
n 5 2
2 1 5 511 5 7(73) 5 (2
2 1)
(2
1 2
1 2
).
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Using these types of patterns, show that
n 5 2
2 1
is composite.
m
c) If m is composite, will the Mersenne number
n 5 2
2 1
always be
composite? Explain.
104
2.3 Factoring Polynomials
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