Question 1209700: Find all roots of the polynomial
f(x) = x^4 - 5x^3 + 5x^2 + 17x - 42 + 4x^4 + 10x^3 - 18x^2 + 2x - 5.
Found 3 solutions by CPhill, mccravyedwin, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! To find the roots of the polynomial f(x) = x^4 - 5x^3 + 5x^2 + 17x - 42 + 4x^4 + 10x^3 - 18x^2 + 2x - 5, we first need to combine the like terms to simplify the polynomial.
Combining the x^4 terms: x^4 + 4x^4 = 5x^4
Combining the x^3 terms: -5x^3 + 10x^3 = 5x^3
Combining the x^2 terms: 5x^2 - 18x^2 = -13x^2
Combining the x terms: 17x + 2x = 19x
Combining the constant terms: -42 - 5 = -47
So, the simplified polynomial is:
f(x) = 5x^4 + 5x^3 - 13x^2 + 19x - 47
To find the roots of this quartic polynomial, we can use numerical methods or a computer algebra system. The roots are approximately:
x ≈ -2.81952163
x ≈ 1.56441331
x ≈ 0.12755416 + 1.45424058i
x ≈ 0.12755416 - 1.45424058i
These are the four roots of the polynomial. Two are real, and two are complex conjugates.
Answer by mccravyedwin(407)  (Show Source):
You can put this solution on YOUR website!
I smell a rat!!! And not just because of the approximate AI solution, allegedly
posted by CPhill(682).
Finding roots of quartics (4th degree polynomial equations) such as this one,
which is solvable only by using Ferrari's method, is a very advanced algebraic
topic which would only be taught in an advanced university course. I studied
both Cardano's method for solving the general cubic, and Ferrari's method for
solving the general quartic as a university junior, in a course which, back
then, was called "Theory of Equations".
So why would a teacher assigning such an advanced problem as this, give the
quartic in the form above, instead of giving it in the form where all the like
terms were already combined, namely, like this:
This is not a quartic polynomial with rational roots that can be solved by
the P/Q method and synthetic division. It can only be solved in terms of
radicals using Ferrari's method.
Teachers of such advanced topics as this DO NOT assign problems in forms that
test them to see if they understand one of the most elementary topics of
beginning algebra, taught in middle school, i.e., that of 'combining like terms'.
This is like a chemistry professor testing chemistry students taking advanced
university chemistry courses, such as qualitative or quantitative analysis, if
they know that the chemical formula for water is H2O.
Thus, I strongly suspect this problem is bogus, posted by a troll.
Edwin
Answer by ikleyn(52794)  (Show Source):
You can put this solution on YOUR website! .
Edwin, let me explain you.
Every time, you think about the students and professors.
It is your paradigm.
Turn the chessboard over.
In this activity, there are no students and professors in the radius of 3000 miles around.
There are business interests, only.
Simply, these guys fill their website/websites with these quasi-"problems".
That is all.
Probably, long time ago this problem did exist in the form as you mentioned
in your post - but now they modified it into this form in their post
to get a "new problem", visually different from the old one.
They do understand nothing in Math, and getting understanding is not their goal.
Their goal is to reach their business interest - nothing more.
Later, they will sell their web-site and will reach money.
From here - their non-sensical problems that flow like a stream, undistinguished one from another.
/////////////////////////////////
Edwin, did you read novels / (short stories) by the American writer O'Henry (1862-1910) ?
He wrote them somewhen the early 20th century.
They all are perfect, first class soft humor.
Especially his stories about Jeff Peters as a personal magnet.
Highly recommend. If not find in the library, try find them online.
About O'Henry, see this Wikipedia article
https://en.wikipedia.org/wiki/O._Henry
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