Question

the graph of $f(x) = 4x^3 - 13x^2 + 9x + 2$ is shown below.
how many roots of $f(x)$ are rational numbers?

Explanation:

Step1: Identify x - intercepts

From the graph, the x - intercepts (where \(y = 0\) or \(f(x)=0\)) are at \(x=\frac{1}{4}\), \(x = 1\), and \(x = 2\). We can visually check the points where the graph crosses the x - axis. The graph crosses the x - axis at three points: one near \(x = 0.25\) (which is \(\frac{1}{4}\)), one at \(x = 1\), and one at \(x=2\).

Step2: Check if they are rational

A rational number is a number that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q
eq0\). \(\frac{1}{4}\) is a fraction of two integers (\(p = 1\), \(q = 4\)), \(1=\frac{1}{1}\), and \(2=\frac{2}{1}\). All three of these x - intercepts are rational numbers. So we count them.

Answer:

3