Quandaries and Queries

your name Stephen
the level of the question secondary (10-12)
who is asking the question Teacher

How do I show kids how to find all the zeros for polynomials degree 4 and bigger.

For examples:
r(x) = x5-11x3-7x2+77 = (x2-11)(x3-7) and s(x) = x4-121 = (x2-11)(x2+11)

Also am I correct when I say that the following are irreducible?

f(x) = x2-11
g(x) = x3-7
h(x) = x4-5

and that a "zero" would be sqrt 11 cuberoot 7 and 4th root of 5

Regards Steve

Hi Steve,

You have set yourself an impossible task. There is no method for finding "all the zeros for polynomials degree 4 and bigger". To be more precise there is no finite expression that only involves only the 4 arithmetic operations (addition, substraction, multiplication and division) and the taking of roots, that will find the zeros of any fifth degree polynomial with integer coefficients. The best you can do is use some appxoximation method to estimate the zeros. For some polynomials, like the two you sent, factorization can reduce the problem to finding the zeros of quadratic or cubic expression.

You are correct in identifying the three expressions as irreducible if you mean irreducible over the integers. You can write

x2-11 = (x - sqrt(11)) (x + sqrt(11)) but there is no polynomial divisor of x2-11 that has only integer coefficients. Hence sqrt(11) and -sqrt(11) are both zeros.

I hope this helps,

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