Questions tagged [solving-polynomials]

For questions about solving polynomials. Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials… like adding, multiplying, polynomial long division, factoring and solving for roots.

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How to work with polynomials in difficult classes?

I am currently teaching a very difficult class, with young people (16 years old) who are deeply unmotivated and restless. I should work with them with polynomials, but I'm having a troublesome time; ...
167 views

How to come up with a Leslie matrix with convenient eigenvalues?

A three by three Leslie matrix looks like $$\begin{bmatrix} f_0 & f_1 & f_2 \\ s_0 & 0 & 0 \\ 0 & s_1 & 0 \end{bmatrix},$$ where $f_0 \ge 0$ and everything else is ...
123 views

What is the notation for polynomial long division in Norway?

I will be teaching a calculus-type course in Norwegian. Our textbook is unfortunately in English (the curse of a small language), but any custom exercises should be and all exams have to be in ...
3k views

Where do students learn to solve polynomial equations these days?

When I was a math undergraduate 30 years ago in India, we were taught what was then called "classical algebra" (as opposed to abstract algebra), and we were taught among other things solving ...
641 views

Is it a problem if a senior student majoring in mathematics could not prove the quadratic formula?

According to a recent experiment conducted by user Steven Gubkin, nearly one half of his students in a senior level Real Analysis course do not have any idea how to prove the quadratic formula. Is ...
2k views

In which course should we teach solving general cubic and quartic equations?

I am guessing solving general cubic and quartic equations should be taught in a course somewhere between precalculus and Galois theory, though personally I do not recall learning this topic ever in ...
2k views

Algebra 2 textbooks that incorrectly claim that all solutions of polynomial equations can be found

Over the years I have occasionally encountered a number of Algebra 2 textbooks that make an incorrect (or at very least extremely misleading) claim along the lines that "all solutions of a polynomial ...
195 views

Beyond cubic polynomials: Applications?

Cubic polynomials are crucially important in computer graphics: for example, cubic Bézier curves/surfaces, and cubic splines, which have many practical applications. Essentially visual continuity ...
710 views

Motivation for polynomial long division

In the U.S. students in grades $\{9,10,11\}$ often learn long division of two polynomials, e.g.: $$\frac{x^4 + 6x^2 + 2}{x^2 + 5} = x^2 + 1 - \frac{3}{x^2 + 5} \;.$$ I believe it is fair to say that ...
323 views

Why is polynomial factorization over the integers part of secondary school curricula?

By "polynomial factorization over the integers", I mean problems and solutions like the following: Problem: Find a factorization into irreducible polynomials for $24x^2 +x - 10$ and ...
2k views

Factoring quadratics where the coefficient on the $x^2$ term does not equal 1

so we are working through various methods of factoring quadratic equations and the students seem comfortable factoring basic quadratics such as: $$x^2 - 7x + 12 = 0$$ by finding the factors of $12$ ...
I usually teach solving linear equations by balancing both sides e.g. $$\begin{array}{cccc} 2x&+&3&=&5\\ &&\color{red}{-3}&&\color{red}{-3}\\ 2x&&&=&2\\... 6answers 905 views Factoring quadratic polynomials In Secondary education in Australia, the general outline for introducing techniques to solve the quadratic equation$$ x^2+bx+c=0  is first to introduce the idea to find two numbers $p$ and $q$ such ...
Eisenstein’s criterion: If you have a polynomial with integer coefficients (and a non-zero constant term) and there’s some prime number $p$ such that $p$ goes into every coefficient but not ...