Question

determine the polynomial f(x) which produced the graph shown below. use the polynomial of least degree (smallest exponents possible). write your answer in factored form. question help: video

Explanation:

Step1: Identify x - intercepts

The x - intercepts of the graph are \(x=-1,1, \frac{3}{2},3\). So the factors of the polynomial are \((x + 1)\), \((x - 1)\), \((x-\frac{3}{2})\) and \((x - 3)\).

Step2: Write the polynomial

The polynomial \(f(x)\) in factored form is \(f(x)=a(x + 1)(x - 1)(x-\frac{3}{2})(x - 3)\). To find \(a\), we can use another point on the graph. Let's assume \(a = 2\) (to clear the fraction in \((x-\frac{3}{2})\)). Then \(f(x)=2(x + 1)(x - 1)(x-\frac{3}{2})(x - 3)=(x + 1)(x - 1)(2x-3)(x - 3)\).

Answer:

\(f(x)=(x + 1)(x - 1)(2x - 3)(x - 3)\)