The expression \[2x^4+ax^3+bx^2-4x-4\] where \(a\) and \(b\) are constants, is denoted by \(f(x)\). Given that \(f\left(-\dfrac{1}{2}\right)=0\) and \(f(2)=0\), find the values of \(a\) and \(b\).

You could use this applet to try some values. The green curve is \(y=f(x)\).

Could we find two equations involving \(a\) and \(b\) that must be simultaneously true?

With these values for \(a\) and \(b\)

  1. express \(f(x)\) as the product of three algebraic factors, and hence show that the equation \(f(x)=0\) has only two real roots;

How would the Factor Theorem help us here?

How can we find the number of real roots of a quadratic equation?

  1. find the set of values of \(x\) for which \(f(x)>0\).

Could sketching the graph of \(f(x)\) help with this?