Further Pure Mathematics WFM01: POLYNOMIALS AND ROOTS Topical Past Paper Questions

$$ \mathrm{f}(x)=x^{2}+qx+r\qquad q,r\in\mathbb{Z} $$ The equation $ f(x)=0 $ has roots $ \alpha $ and $ \beta $ (a) Write down, in terms of $q$ or $r$ (i) $$ \boldsymbol{\alpha}+\boldsymbol{\beta} $$ $$ \begin{aligned}&\because\alpha\parallel\beta\\ &=\alpha+\beta\\ &=\alpha+\be…

$$ \mathrm{f}(z)=P z^{4}-36z^{3}+Q z^{2}+192z+68 $$ where P and Q are real constants. Given that $3 + 5i$ is a root of the equation $f(z) = 0$ (a) write down another root of the equation $ f(z) = 0 $ (1) (b) Hence determine a quadratic factor, with real coefficients, of $f(z)$ (2…

$$ 4x^{2}+4x+17\equiv(2x+p)^{2}+q $$ where $p$ and $q$ are integers. (a) Determine the value of p and the value of q (2) Given that $$ \frac{8x+5}{\sqrt{4x^{2}+4x+17}}\equiv\frac{1}{\sqrt{4x^{2}+4x+17}}+\frac{Ax+B}{\sqrt{4x^{2}+4x+17}} $$ where $A$ and $B$ are integers, (b) write…