305 results for "polynomial".

Question in Howtos
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.

Question in Content created by Newcastle University
Adding linear combinations of polynomials over $\mathbb{Z}_3,\;\mathbb{Z}_5,\;\mathbb{Z}_7$

Question in Content created by Newcastle University
3 Repeated integrals of the form $\int_a^b\;dx\;\int_c^{f(x)}g(x,y)\;dy$ where $g(x,y)$ is a polynomial in $x,\;y$ and $f(x)$ is a degree 0, 1 or 2 polynomial in $x$.

Question in Content created by Newcastle University
Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

Question in Content created by Newcastle University
Find the first 3 terms in the MacLaurin series for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.

Question in Content created by Newcastle University
Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$ with coefficients in the real numbers. Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by: \[\phi(p(x))=p(a)+p(bx+c).\]Using the standard basis for range and domain find the matrix given by $\phi$.

Question in Content created by Newcastle University
Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$ with coefficients in the real numbers.
Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by:
$\phi(p(x))=ap(x) + (bx + c)p'(x) + (x ^ 2 + dx + f)p''(x)$
Using the standard basis for range and domain find the matrix given by $\phi$.

Question in Content created by Newcastle University
$\displaystyle \int \frac{bx+c}{(ax+d)^n} dx=g(x)(ax+d)^{1n}+C$ for a polynomial $g(x)$. Find $g(x)$.

Question in Content created by Newcastle University
Find $\displaystyle\int \frac{ax^3ax+b}{1x^2}\;dx$. Input constant of integration as $C$.

Question in Content created by Newcastle University
Find $\displaystyle \int \frac{nx^3+mx^2+nx + p}{1+x^2}\;dx$. Solution involves $\arctan$.

Question in Content created by Newcastle University
Find $\displaystyle \int \frac{nx^3+mx^2+px +m}{x^2+1} \;dx$

Question in Content created by Newcastle University
Find the polynomial $g(x)$ such that $\displaystyle \int \frac{ax+b}{(cx+d)^{n}} dx=\frac{g(x)}{(cx+d)^{n1}}+C$.

Question in Content created by Newcastle University
Find $\displaystyle \int\frac{ax^3+ax+b}{1+x^2}\;dx$. Enter the constant of integration as $C$.

Question in Content created by Newcastle University
Find $\displaystyle \int \frac{nx^3+mx^2+nx + p}{1+x^2}\;dx$. Solution involves $\arctan$.

Question in Content created by Newcastle University
Given that $\displaystyle \int x({ax+b)^{m}} dx=\frac{1}{A}(ax+b)^{m+1}g(x)+C$ for a given integer $A$ and polynomial $g(x)$, find $g(x)$.

Question in Content created by Newcastle University
3 Repeated integrals of the form $\int_a^b\;dx\;\int_c^{f(x)}g(x,y)\;dy$ where $g(x,y)$ is a polynomial in $x,\;y$ and $f(x)$ is a degree 0, 1 or 2 polynomial in $x$.

Question in Content created by Newcastle University
The derivative of $\displaystyle \frac{ax+b}{\sqrt{cx+d}}$ is $\displaystyle \frac{g(x)}{2(cx+d)^{3/2}}$. Find $g(x)$.

Question in Content created by Newcastle University
The derivative of $\displaystyle \frac{ax^2+b}{cx^2+d}$ is $\displaystyle \frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.

Question in Content created by Newcastle University
The derivative of $\displaystyle \frac{ax+b}{cx^2+d}$ is of the form $\displaystyle \frac{g(x)}{(cx^2+d)^2}$. Find $g(x)$.

Question in Content created by Newcastle University
Differentiate $\displaystyle e^{ax^{m} +bx^2+c}$

Question in Content created by Newcastle University
Differentiate
\[ \sqrt{a x^m+b})\]

Question in Content created by Newcastle University
Using a given list of four complex numbers, find by inspection the one that is a root of a given cubic real polynomial and hence find the other roots.

Question in Content created by Newcastle University
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.

Question in Content created by Newcastle University
Find the first 3 terms in the Taylor series at $x=c$ for $f(x)=(a+bx)^{1/n}$ i.e. up to and including terms in $x^2$.

Exam (13 questions) in Content created by Newcastle University
Questions about complex arithmetic; argument and modulus of complex numbers; complex roots of polynomials; de Moivre's theorem.

Question in Content created by Newcastle University
Finding the coordinates and determining the nature of the stationary points on a polynomial function

Question in Content created by Newcastle University
Complete the square for a quadratic polynomial $q(x)$ by writing it in the form $a(x+b)^2+c$. Find both roots of the equation $q(x)=0$.

Question in Content created by Newcastle University
Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.

Question in Content created by Newcastle University
$f(X)$ and $g(X)$ as polynomials over the rational numbers $\mathbb{Q}$.
Find their greatest common divisor (GCD) and enter as a normalized polynomial.

Question in Content created by Newcastle University
Expanding products of 3 linear polynomials over $\mathbb{Z}_3,\;\mathbb{Z}_5,\;\mathbb{Z}_7$