notes / Integrated Mathematics / chapter 10: Polynomials
10.11 Polynomial Inequalities
To solve a polynomial inequality, find the roots, factor, and check the sign of each factor on the intervals between the roots.
Problem 10.11.1. Solve .
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are the possible rational roots.
. Given , and for the roots , either roots are negative and positive, or all roots are positive.
Let’s try , and : . YAY!
So .
Problem 10.11.2. Find all such that .
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Let’s first find the roots (), trying integers first.
: ; : .
: and , so . So and , so .
Try :
Candidates: under , i.e. :
So we need
Roots , , :
Exercises
Exercise 10.11.3. (a)
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so the inequality is .
(b)
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where :
, so
, so
Testing signs: between and , ; below , ; at , . So
(c)
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Possible roots: . , so
and has roots
so