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Question 7317: solve the following equation by finding its zeros.
x^3+9x^2+26x+24=0
Answer by glabow(165)   (Show Source):
You can put this solution on YOUR website! The best way to go about this kind of problem is to analyze the equation. The integer root theorem says that any root of a polynomial that has integer coefficients and a leading coefficient of 1 must be a factor of the constant term.
The leading coefficient of is 1, all the other coefficients are integers, so any root must be factor of 24. The factors of 24 are +- (1, 2, 3, 4, 6, 8, 12, 24).
Here's where skill and guesswork come in. You try each of these in the polynomial and see if they result in a value of 0!
Actually, you might see that if the root you guess is positive, and all the terms are positive, it will never be zero since each term just adds to the value. So, you would only try negative factors.
The first one I'd try is -3, and it's just a guess.
[This next part uses synthetic substitution. If you don't know that yet, look it up and learn it. It will save you LOTS of work.]
Synthetic division gives the following:
......-3.|.1.....9.....26.....24
............|......-3....-18...-24
-------------------------------------
...............1.....6.......8........0
With a remainder of 0 we see that -3 is a root. That means that (x + 3) is a factor of the polynomial. And the intermediate results of the synthetic substitution are the coefficients of the reduced polynomial. So,
Fortunately, the reduced polynomial is easy to factor. Actually, this frequently happens. If it weren't, you could use the integer root theorem again on this polynomial. It shows the factors must be from the set +- (1, 2, 4, 8). But two numbers that multiply to 8 and add up to 6 are easy to imagine.
 shows the roots to be -2, -3, and -4.
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