Polynomials

What is the degree of $(x^3 + 2x)(4x^5 - x + 1)$?

Find the remainder when $p(x) = x^3 - 4x^2 + 2x + 7$ is divided by $x - 3$.

For which number $k$ is $x - 2$ a factor of $x^3 + kx - 6$?

Find all real roots of $x^3 - 2x^2 - 5x + 6$. Enter them as a set, for example {1, 2, 3}.

For $x \ne 0$, which expression equals $\dfrac{x^2 + 1}{x}$?

A polynomial with real coefficients has degree $4$. How many different real roots could it have? Choose all that are possible.

In the proof of the root bound we wrote: $0 = p(b) = (b - a)\,q(b)$ and $b - a \ne 0$, so $q(b) = 0$. Modulo $8$ this step fails and $x^2 - 1$ has four roots. Which property of the real numbers does the step use?

Which Lean statement formalises the factor theorem for a polynomial p : R[X] and a number a : R? (Recall: C a is the constant polynomial $a$ and p.eval a is $p(a)$.)

Write divide_by_linear(coeffs, a) that divides $p(x)$ by $x - a$ using synthetic division. coeffs lists the coefficients of $p$ from the highest power down (so $2x^3 - 3x^2 - 11x + 6$ is [2, -3, -11, 6]). Return a pair (quotient, remainder), where quotient is the list of the quotient's coefficients.