Polynomial equations with the solver Problem set A

Solve \(x^3-2x^2=5x-3\).

Find the real roots of \(x^4-6x^2=2x-3\).

Find the real root of \(2x^3+3x+8=0\).

Find the real root of \(x^3+x^2=-5x-9\).

Given that \(2x+\frac{3}{x}=7\):

1. (a) show that \(2x^2 - 7x + 3=0\);
2. (b) hence find the possible values of x.

A block of wood is a cuboid measuring x cm by \((x+4)\) cm by \((x+7)\) cm. Its volume is 500 cm³.

1. (a) Show that \(x^3 + 11x^2 + 28x - 500=0\).
2. (b) Find the value of x.

A train travels 240 km at a steady speed of v km/h. A faster train travels the same 240 km at \((v+20)\) km/h and takes 1 hour less, so \(\frac{240}{v}-\frac{240}{v+20}=1\).

1. (a) Show that \(v^2+20v-4800=0\).
2. (b) Find the speed of the slower train.

Given that \(2x-\frac{3}{x+1}=4\):

1. (a) show that \(2x^2 - 2x - 7=0\);
2. (b) hence find the possible values of x.

The number of bacteria in a culture, in thousands, is modelled by \(N=t^3-6t^2+14t\), where t is the time in hours after the start. Find when the culture first reaches 40 thousand bacteria.