Solve this equation for R

Question

4R³-499R-1575=0

I came across this equation as part of the solution of another problem elsewhere.

It is an example of casus irreducibilis. However, it has a rational solution which can be obtained by factoring.

I know the answer, but what I really would like to know is how you arrive at the mechanism for factoring, i.e. how do you come up with the manipulations which produce an easily factorable cubic.

Answer 1

There is a method for solving cubics (similar to the quadratic formula) BUT it is very complicated. I built a little Excel file that does it - and it works - but like I said, it's super complicated.

Normally, I would say that for something like the equation you listed, use the Rational Roots Theorem and synthetic division.

IF there is a rational solution, it will be one of the factors of 1575 divided by one of the factors of 4. The factors of 1575 are:

1, 3, 5, 7, 9, 15, 21, 25, 35, 45, 63, 75, 105, 175, 225, 315, 525, and 1575 and the factors of 4 are 1, 2, and 4 so plus or minus any of the various combinations.

Start easy, with 1. Using synthetic division, we get a remainder of -2070 so (1, -2070) is on the curve.

Try -1, using synthetic division, we get a remainder of -1080 so (-1, -1080) is on the curve.

It looks like we're getting closer to a zero, so go a little more negative: Try -3 and we get a remainder of -186 so (-3, -186) is on the curve so it appears we are getting closer to a zero.

Try -5: using synthetic division, we get a remainder of 420 so (-5, 420) is on the curve meaning we passed the zero. Looking at the possible combinations, I see that I can get -3.5 by dividing 7 and 2. So try -3.5: Using synthetic division, we get a remainder of 0 so x = -3.5 is a root. The synthetic division reveals that the quotient when dividing by (R + 3.5) is 4R² -14R - 450. We can either factor this using regular quadratic trinomial methods or use the quadratic formula. Since it's still fairly complicated, I used the quadratic formula and got R = 12.5 and R = -9 resulting in the following factorization:

R³ - 499R - 1575 = 4(R + 3.5)(R - 12.5)(R + 9)

Breaking the leading coefficient into two "2's" and incorporating a 2 into each of the "decimal" terms, we get:

R³ - 499R - 1575 = (2R + 7)(2R - 25)(R + 9)

Answer 2

First, the Rational Root Theorem

Factors of 1575 are (plus or minus) 1, 3, 5, 7, 9, 15, 21, 25, 35, 45, 63, 75, 105, 175, 225, 315, 525, 1575

Factor of 4 are (plus or minus) 1, 2, 4

List the possible rational roots: Factors of 1575 / Factors of 4

Second, Synthetic Substitution to determine a root.

Third, Factor depressed equation to determine remaining two roots.

Answer 3

Thank you, Mark & WIlliam

Your answers are both obviously correct...and since one of the 3 rational roots is -9 you might fairly easily find it by your methods. However, you didn't really answer my question.

The factoring solution is as follows:

4R³-(18²+175)R-9*175=0

4R(R²-9)-175(R+9)=0...and you can go on from there easily.

The question is how would you teach someone (myself or a student) to find that manipulation of coefficients to get the factored solution. I looked at the equation for a very long time and didn't see it!

Any thoughts?