If \(f(x)=0\) is an algebraic equation of integral degree, show that the sum of the \(m\)th powers of its roots is the coefficient of \(x^{-m}\) in the expansion of \(xf'(x)/f(x)\). Find the sum of the cubes of the roots of the quartic equation \[ x^4+x^3-2x^2-7=0. \]
Solution: Notice that \begin{align*} \frac{xf'(x)}{f(x)} &= x \cdot \frac{\left ( \prod (x-\alpha_i)^{k_i} \right)'}{\prod (x-\alpha_i)^{k_i}} \\ &= x \cdot \frac{\sum_i \prod_{j \neq i} (x-\alpha_j)^{k_j} k_i(x-\alpha_i)^{k_i-1}}{\prod (x-\alpha_i)^{k_i}} \\ &= x \sum_i \frac{k_i}{(x-\alpha_i)} \\ &= \sum_i \frac{k_i}{1 - \frac{\alpha_i}{x}} \\ &= \sum_i \sum_{m=0}^\infty k_i \left ( \frac{\alpha_i}{x} \right)^m \\ &= \sum_{m=0}^\infty \underbrace{\left ( \sum_{i} k_i \alpha_i^m \right)}_{\text{sum of the }m{\text{th powers of roots}}} x^{-m} \end{align*} We are looking for the coefficient of \(x^{-3}\) in the expansion of: \begin{align*} \frac{xf'(x)}{f(x)} &= \frac{x(4x^3+3x^2-4x)}{ x^4+x^3-2x^2-7} \\ &= \frac{4x^4+3x^3-4x^2}{x^4+x^3-2x^2-7} \\ &= \frac{4+3x^{-1}-4x^{-2}}{1+x^{-1}-2x^{-2}-7x^{-4}} \\ &= (4+3x^{-1}-4x^{-2})(1 -(x^{-1}-2x^{-2}-7x^{-4})+(x^{-1}-2x^{-2}-7x^{-4})^2-(x^{-1}-2x^{-2}-7x^{-4})^3 + \cdots) \\ &= (4+3x^{-1}-4x^{-2})(1 -(x^{-1}-2x^{-2})+(x^{-1}-2x^{-2})^2-(x^{-1})^3) + o(x^{-4}) \\ &= (4+3x^{-1}-4x^{-2})(1 -x^{-1}+2x^{-2}+x^{-2}-4x^{-3}-x^{-3}) + o(x^{-4})\\ &= (4+3x^{-1}-4x^{-2})(1 -x^{-1}+3x^{-2}-5x^{-3}) + o(x^{-4})\\ &= \cdots + (4+9-20)x^{-3} + \cdots \end{align*} Threfore the sum of the cubes is \(-7\).
Prove that the arithmetic mean of a set of unequal positive quantities is greater than their geometric mean. Hence establish that if \(n\) is a positive integer
Explain briefly the rule for multiplying two determinants together. Show that \[ \begin{vmatrix} b^2+c^2+1 & c^2+1 & b^2+1 & b+c \\ c^2+1 & c^2+a^2+1 & a^2+1 & c+a \\ b^2+1 & a^2+1 & a^2+b^2+1 & a+b \\ b+c & c+a & a+b & 3 \end{vmatrix} \] is the square of a certain determinant, and hence obtain its value.
Prove that if for a polynomial \(f(x)\) of degree \(n\) with real coefficients the values of \(f(x)\) and all its derivatives are positive for \(x=x_0\), then no root of \(f(x)=0\) can exceed \(x_0\). Prove also the truth of this statement for the equations \(f^{(r)}(x)=0\) obtained by differentiating \(f(x)\) \(r\) times. Consider the case \(f(x)=x^4-2x^3-3x^2-15x-3\) and show that all the roots of \(f(x)=0\) are less than 4.
The polynomial \(f_n(x)\) is defined as \(\dfrac{d^n}{dx^n}(x^2-1)^n\). Prove that all the roots of the equation \(f_n(x)=0\) are real and distinct and lie between \(\pm 1\). Prove also that \(\int_{-1}^1 f_n(x)f_m(x)dx=0\) if \(m \neq n\), and find its value when \(m=n\).
Define \(\log_e x\) for \(x>0\). Prove that for \(x>1\): \[ x^2-x > x\log_e x > x-1 \quad \text{and} \quad x^2-1 > 2x\log_e x > 4(x-1)-2\log_e x. \]
Evaluate the following integrals: \[ \int_{\pi/4}^{3\pi/4} \frac{dx}{2\cos^2 x+1}; \quad \int_0^\infty \frac{dx}{(1+x^2)^n}, \text{where } n \text{ is a positive integer}; \quad \int_0^\infty \frac{dx}{1+x^3}. \]
Discuss the general nature of the plane curve whose polar equation is \(r = \dfrac{a}{\theta^2-1}\) for values of \(\theta>1\). Prove that one of the bisectors of the angle between the radius vector and the normal is inclined at an angle \(\tan^{-1}\theta\) to the radius vector, and find an expression for the length of arc from the origin to the given point.
A sphere of radius \(a\) has centre \(O\), and \(P\) is a point distant \(z\) from \(O\). Find the mean value with respect to area of the \(n\)th power of the distance of the surface of the sphere from \(P\), where \(n \ge -1\) and is not necessarily integral, distinguishing between the cases when \(z > a\) and \(z < a\). Verify that if \(P\) is external to the sphere and \(P'\) is the inverse of \(P\) with respect to the sphere, the mean value for \(P'\) is \((\frac{a}{z})^n\) that for \(P\).
A tripod consists of three uniform rods \(AB, AC\) and \(AD\), each of length \(l\) and weight \(W\), smoothly jointed at \(A\). It rests in the form of a regular tetrahedron, with apex \(A\), upon a smooth horizontal surface. The feet \(B\) and \(C\) are fixed (the rods \(AB\) and \(AC\) being free to rotate about these points), and the foot \(D\) is prevented from slipping by inextensible strings \(BD\) and \(CD\). A horizontal force \(F\), in the direction of the perpendicular from \(D\) to \(BC\), acts at \(A\). Calculate (i) the force of interaction between \(AD\) and the surface, (ii) the tension in the strings. If the magnitude of the applied force is gradually increased, for what value of \(F\) will equilibrium be broken?