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kkaman
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If 1/a^3+ a^2+ 9=0 , is ''a'' greater than/less than or equal to -1/3.
This a GRE question. Thanks!
This a GRE question. Thanks!
He actually multiplied by a^3, but wrote a^2.jacobrhcp said:I think you may have missed the '1/' at the beginning, Halls. that one does not make it more cubic, though.
HallsofIvy said:That doesn't look like a cubic equation to me. If you multiply both sides by a3 you get a5+ 9a3+ 1= 0, a fifth degee equation. We can tell by "DesCartes' rule of signs" that it has no positive real root and only one negative real root. When a= 0, (0)5+ 9(0)+ 1= 1 which is positive and when a= -1/3, (-1/3)5+ 9(-1/3)+ 1= -2+ 1/243 which is negative. What does that tell you?
A cubic equation is a polynomial equation of the form ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants and x is the variable.
There are multiple methods for solving a cubic equation, including the rational root theorem, completing the square, and using the cubic formula. The most common method is to use the cubic formula, which involves finding the roots using a quadratic formula.
A cubic equation can have up to three solutions, which can be real or complex. However, there is always at least one real solution for a cubic equation.
The coefficients of a cubic equation can provide information about the solutions. For example, the sum of the solutions is equal to -b/a, and the product of the solutions is equal to -d/a.
Yes, a cubic equation can have irrational solutions. For example, the equation x^3 - 3x + 1 = 0 has one real solution that is irrational (approximately 1.32472).