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You are given a polynomial f, where f(x)=x^4 - 14x^3 + 74 x^2 -184x + 208, you are told that f(5+i)=0. Express f as the product of two quadratic polynomials and state all roots of f.

Since x=5+i is a solution to f(x)=0 we then know that x=5-i must also be a solution to f(x)=0, by the complex conjugate root theorem.Now we can break f down into the product of a polynomial and these two kno...
PL
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How do I find and plot the roots of a polynomial with complex roots on an Argand diagram? e.g. f(z) =z^3 -3z^2 + z + 5 where one of the roots is known to be 2+i

For a polynomial with real coefficients, use that roots come in complex conjugate pairs. Therefore, another root is 2-i (and we know for this example that the final root must be real). Write the factorised f...
ES
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Using a Suitable substitution or otherwise, find the differential of y= arctan(sinxcosx), in terms of y and x.

First of all, replace sinxcosx with 1/2 sin2x. Then you should let U=1/2 Sin2x and replace that in the formula. If y=arctan(U), then U=tany. work out dU/dy which is Sec 2 y. Using the trigonometric identity ...
JP
3021 Views

Prove by induction that, for all integers n >=1 , ∑(from r=1 to n) r(2r−1)(3r−1)=(n/6)(n+1)(9n^2 -n−2). Assume that 9(k+1)^2 -(k+1)-2=9k^2 +17k+6

First, recall how to construct a proof by simple induction in this manner: (1) Assume statement true for n=k, (2) Prove true for n=k+1, (3) Show true for n=1.(1) => ∑(from r=1 to k) r(2r−1)(3r−1)= (k/6)(k...
TK
8064 Views

A spring with a spring constant k is connected to the ceiling. First a weight of mass m is connected to the spring. Deduce the new equilibrium position of the spring, find its equation of motion and hence deduce its frequency f.

Let's first find the equilibrium position of the spring. When the mass is first attached the spring will oscillate like a simple harmonic oscillator, in the real world the oscillator will eventually settle a...
BP
3117 Views