5 Questions
Between x = 1/a and x = 4/c, the sign of the polynomial will ______
alternate
In the interval (-∞, 0), the sign of the polynomial is ______
negative
In the interval (0, 1/a), the sign of the polynomial is ______
positive
In the interval (1/a, 4/c), the sign of the polynomial is ______
positive
In the interval (4/c, ∞), the sign of the polynomial is ______
negative
Study Notes
Solving Polynomial Inequality
- The polynomial inequality is x(ax-1)^2(cx-4) > 0, with roots x = 0, x = 1/a, and x = 4/c.
- The root x = 1/a is a double root, which means the sign of the polynomial will not change at x = 1/a.
- The sign of the polynomial will alternate only between x = 1/a and x = 4/c.
Intervals and Test Points
- Divide the number line into intervals: (-∞, 0), (0, 1/a), (1/a, 4/c), and (4/c, ∞).
- Choose test points for each interval: -1, 1/2, 3, and 5, respectively.
Sign of the Polynomial
- For x < 0, the polynomial is negative: x < 0, (ax-1)^2 > 0, and cx-4 < 0.
- For 0 < x < 1/a, the polynomial is positive: x > 0, (ax-1)^2 > 0, and cx-4 < 0.
- For 1/a < x < 4/c, the polynomial is positive: x > 0, (ax-1)^2 > 0, and cx-4 > 0.
- For x > 4/c, the polynomial is negative (not explicitly stated in the text, but follows from the pattern).
Inequality of Polynomial Roots Quiz: Test your understanding of polynomial inequalities by solving for the sign changes of a given polynomial expression with multiple roots. Explore how to determine the sign of the polynomial in different intervals using test points.
Make Your Own Quizzes and Flashcards
Convert your notes into interactive study material.
Get started for free
