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The euclidean algorithm in the polynomial ring $\mathbb Q[x]$.

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Practice dividing polynomials using the long division method.

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The data is fitted by linear and quadratic regression.  First, find a linear regression equation for the $n$ data points, $20 \le n \le 35$.

They then are shown that the quadratic regression is often a better fit as measured by SSE. Also users can experiment with fitting polynomials of higher degree.

Factorise polynomials by identifying common factors. The first expression has a constant common factor; the rest have common factors involving variables.

Differentiation of  polynomials, cos, sin, exp, log functions. Product, quotient and chain rules.

rebelmaths

The student's answer is a fraction of two polynomials. First check that the student's answer is a fraction, then check that the numerator is of the form $x+a$.

To find the script, look in the Scripts tab of part a.

Factorising polynomials using the highest common factor

Finding the zero of a non-polynomial function

Differentiate

\[ \sqrt{a x^m+b})\]

This question tests the student's knowledge of the remainder theorem and the ways in which it can be applied.

Quotient and remainder, polynomial division.

The derivative of $\displaystyle \frac{ax^2+b}{cx^2+d}$ is $\displaystyle \frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.

Contains a video solving a similar quotient rule example. Although does not explicitly find $g(x)$ as asked in the question, but this is obvious.

Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.

Divide $ f(x)=x ^ 4 + ax ^ 3 + bx^2 + cx+d$ by $g(x)=x^2+p $ so that:
$\displaystyle \frac{f(x)}{g(x)}=q(x)+\frac{r(x)}{g(x)}$

Find the polynomial $g(x)$ such that $\displaystyle \int \frac{ax+b}{(cx+d)^{n}} dx=\frac{g(x)}{(cx+d)^{n-1}}+C$.

Algebraic manipulation/simplification.

Simplify $\displaystyle \frac{ax^4+bx^2+c}{a_1x^4+b_1x^2+c_1}$ by cancelling a a common degree 2 factor.

Differentiate $ (ax+b)^m(cx+d)^n$ using the product rule. The answer will be of the form $(ax+b)^{m-1}(cx+d)^{n-1}g(x)$ for a polynomial $g(x)$. Find $g(x)$.

Differentiate $ x ^ m(ax+b)^n$ using the product rule. The answer will be of the form $x^{m-1}(ax+b)^{n-1}g(x)$ for a polynomial $g(x)$. Find $g(x)$.

The derivative of  $\displaystyle \frac{ax+b}{\sqrt{cx+d}}$ is $\displaystyle \frac{g(x)}{2(cx+d)^{3/2}}$. Find $g(x)$.

The derivative of $\displaystyle \frac{ax+b}{cx^2+dx+f}$ is $\displaystyle \frac{g(x)}{(cx^2+dx+f)^2}$. Find $g(x)$.

Differentiate $ x ^m \sqrt{a x+b}$.
The answer is in the form $\displaystyle \frac{x^{m-1}g(x)}{2\sqrt{ax+b}}$
for a polynomial $g(x)$. Find $g(x)$.

A basic introduction to differentiation

Given two complex numbers, find by inspection the one that is a root of a given quartic real polynomial and hence find the other roots. 

This question provides practice at differentiating polynomials.

Multiple choice question. Given a randomised polynomial select the possible ways of writing the domain of the function.

Multiple choice question. Given a randomised polynomial select the possibe ways of writing the domain of the function.