'Polynomial Inequalities' Algebra Worksheet
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1
2POLYNOMIAL INEQUALITIES
_______________________
Name:
Section: ___________
Checked by: _________________________
____________________
_________________________
Course Objective: Solve absolute value, polynomial, and rational inequalities
Scoring: Your work and answer will be checked on 5 randomly picked questions. Neatness may count for several points.
In calculus, a method of solving factorable polynomial inequalities may be used 3 times in one exercise as follows:
1.
Solve a polynomial inequality to determine where a graph is above/below the x-axis
2.
Find the derivative of the polynomial function (requires easy calculus)
3.
Solve the resulting polynomial inequality to determine where the graph is increasing or decreasing.
4.
Find the next derivative (more easy calculus)
5.
Solve the resulting polynomial inequality to determine how the graph curves
Example:
2
2
a) Use the graph to solve
x
− x
2
−
3
<
0
;
b) Use the graph to solve
x
− x
2
−
3
≥
0
20
SOLUTION:
15
2
a)The graph of
y
=
x
−
2
x
−
3
is below the x-axis (
0
) only for x-values strictly
y
<
10
between -1 and 3,
(−
1
) 3 ,
.
5
2
b) The graph of
y
=
x
−
2
x
−
3
is above the x-axis(
y
≥
0
) for x-values less than or
4
2
2
4
6
equal to -1,or for x-values greater than or equal to 3.
(
−∞
,
−
] 1
∪
, 3 [
∞
)
To develop an algebraic method of solving inequalities, we look at factors and how they work together.
Recall: Let P and Q be factors of a polynomial.
then
PQ
<
0
if
P
<
0
or
Q
<
0
but not both;
and
PQ
>
0
if
P
<
0
and
Q
<
0
or if
P
>
0
and
Q
>
0
It is important to determine x-values at which P and Q change signs (from positive to negative or from negative to positive).
We will call these values key numbers.
2
METHOD: to solve
x
− x
2
−
3
≥
0
1.
Factor the polynomial,
(
x
−
3
)(
x
+
) 1
≥
0
.
2.
Find the key numbers. These are the numbers for which each factor equals zero. Here:
x
=
3
and
x
=
1 −
.
3.
List factors in a column. Below and to the right, draw a number line and label key numbers.
(
)
x
+
1
(
x
−
) 3
___________|_____________________|_________________
-1
3
4. Test each factor in the three regions determined.
(x + 1)
--
+
+
(x – 3)
--
--
+
___________|_____________________|_________________
+
-1
--
3
+
5. “Multiply” signs and place resultant sign below number line.
2
6. Conclude:
x
− x
2
−
3
≥
0
for x-values in
(
−∞
,
−
] 1
∪
, 3 [
∞
)
(intervals for which the resultant sign is +, i.e. > 0)
7. Ideas extend for more than two factors.
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