1.2.10 Manipulating polynomials

Let $f(x) = 36x^3 - 27x^2 - 4x + 3$ .

Use the factor theorem to show that $(3 x - 1)$ is a factor of $f (x)$ .

[2]

Express $f (x)$ as a product of three linear factors.

[3]

A mechanical engineer modeling the resonance of a suspension bridge determines that the equilibrium angles $θ\theta$ are given by the equation:

\frac{12\sec^2 \theta + \cos \theta}{4} = 2.25\sec \theta + \frac{1}{3}

Prove that there are no real values of $θ\theta$ that satisfy this stability condition.

[4]

1.2.10 Manipulating polynomials Questions

Let $f(x) = 36x^3 - 27x^2 - 4x + 3$ .

Use the factor theorem to show that $(3 x - 1)$ is a factor of $f (x)$ .

[2]

Express $f (x)$ as a product of three linear factors.

[3]

A mechanical engineer modeling the resonance of a suspension bridge determines that the equilibrium angles $θ\theta$ are given by the equation:

\frac{12\sec^2 \theta + \cos \theta}{4} = 2.25\sec \theta + \frac{1}{3}

Prove that there are no real values of $θ\theta$ that satisfy this stability condition.

[4]