Three Polynomial Questions
Date: 7 Aug 1995 09:00:07 -0400 From: Greg Sharpe Subject: Polynomial Questions (3) Hi! I have 3 interesting polynomial questions from a past trial HSC paper. I am in year 12 in NSW, Australia. Are there any 'intereactive' real time discussion groups for maths problems? Currently, I have www and email access. Thanks in advance, Greg Sharpe <firstname.lastname@example.org> 1) The polynomial 2x^3 + 3x^2 + ax - 6 has (x + 3) and (2x + b) as factors. Find a and b. 2) In the polynomial x^3 - 4x^2 + Ax + 4 = 0, one root is equal to the sum of the other two roots. Solve the equation and find A. 3) Find the roots of 2x^3 - 3x^2 - 3x + 2 = 0 given that they are in geometric progression.
Date: 7 Aug 1995 09:44:23 -0400 From: Dr. Ken Subject: Re: Polynomial Questions (3) Hello there! Nice questions. >1) The polynomial 2x^3 + 3x^2 + ax - 6 has (x + 3) and (2x + b) as factors. > Find a and b. To do this one, try doing polynomial long division, dividing (x + 3) into the cubic polynomial. Since it goes in evenly without remainder, that should tell you what a is. Then once you've done that, you'll have a quadratic polynomial, and you can divide (2x + b) into it to find out what b is, or you could use the quadratic formula to find out what the roots are and then use that to figure out what b is. >2) In the polynomial x^3 - 4x^2 + Ax + 4 = 0, one root is equal to > the sum of the other two roots. Solve the equation and find A. To figure this out, you'll most likely have to use the fact that the coefficient on x^2 is the opposite of the sum of the roots. So if the roots are p, q, and r, then you have the equation p + q + r = 4. Also, the constant term in the polynomial is the opposite of the product of the roots, so we have the equation pqr = -4. Since you said that one root is the sum of the other two, we have the equation p + q = r. That's three equations and three unknowns. That should do it for you! (especially look at the first and the last equation together) >3) Find the roots of 2x^3 - 3x^2 - 3x + 2 = 0 given that they are in > geometric progression. Let's call a, ar, and ar^2 the three roots, with common ratio r. Then expand the polynomial 2(x - a)(x - ar)(x - ar^2) and see how that expansion matches up with the given polynomial. Good luck! -K
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