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Questions and Answers
Between x = 1/a and x = 4/c, the sign of the polynomial will ______
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 ______
In the interval (-∞, 0), the sign of the polynomial is ______
negative
In the interval (0, 1/a), the sign of the polynomial is ______
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 ______
In the interval (1/a, 4/c), the sign of the polynomial is ______
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In the interval (4/c, ∞), the sign of the polynomial is ______
In the interval (4/c, ∞), the sign of the polynomial is ______
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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).
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