Factoring Worksheet With Examples Page 26
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50A polynomial that cannot be written as a product of two polynomials with
integral coefficients is called a prime polynomial.
EXAMPLE
Determine Whether a Polynomial Is Prime
2
+ 5x - 2.
Factor 2 x
In this trinomial, a = 2, b = 5, and c = -2. Since b is positive, m + n is
positive. Since c is negative, mn is negative. So either m or n is negative, but
not both. Therefore, make a list of the factors of 2(-2) or -4, where one
factor in each pair is negative. Look for a pair of factors with a sum of 5.
Factors of -4
Sum of Factors
1, -4
-3
-1, 4
3
-2, 2
0
2
There are no factors with a sum of 5. Therefore, 2 x
+ 5x - 2 cannot be
2
factored using integers. Thus, 2 x
+ 5x - 2 is a prime polynomial.
2
2
2
A.
Is 4 r
- r + 7 prime?
2
B.
Is 2 x
+ 3x - 5 prime?
Solve Equations by Factoring
2
Some equations of the form a x
+ bx + c = 0
can be solved by factoring and then using the Zero Product Property.
EXAMPLE
Solve Equations by Factoring
2
- 9a - 5 = 4 - 3a. Check the solutions.
Solve 8 a
2
8 a
- 9a - 5 = 4 - 3a
Write the equation.
2
8 a
- 6a - 9 = 0
Rewrite so that one side equals 0.
(4a + 3)(2a - 3) = 0
Factor the left side.
4a + 3 = 0
or
2a - 3 = 0
Zero Product Property
4a = -3
2a = 3
Solve each equation.
_
_
3
3
a = -
a =
4
2
_
_
3
3
The roots are -
and
.
4
2
CHECK
Check each solution in the original equation.
2
2
8
a
-
9a
- 5 = 4 -
3a
8
a
-
9a
- 5 = 4 -
3a
_
_
_
_
_
_
2
2
(
)
(
)
(
)
(
)
(
)
(
)
3
3
3
3
3
3
8
-
- 9
-
- 5
4 - 3
-
8
- 9
- 5
4 - 3
4
4
4
2
2
2
_
_
_
_
_
27
9
9
27
9
18 -
- 5
4 -
+
- 5
4 +
2
2
2
4
4
_
_
_
_
1
1
25
25
-
= -
=
2
2
4
4
2
2
3
A.
3 x
- 5x = 12
3
B.
2 x
- 30x + 88 = 0
2
443
Extra Examples at
Lesson 8-4 Factoring Trinomials a x
+ bx + c
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