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:

Find potential rational roots

Using the Rational Root Theorem knowledge point

$$ \text{Possible rational roots of } f(x) = 4x^3 - 13x^2 + 9x + 2 \text{ are } \pm \frac{p}{q} $$

where \(p\) divides \(2\) (factors: \(\pm 1, \pm 2\)) and \(q\) divides \(4\) (factors: \(\pm 1, \pm 2, \pm 4\)).

$$ \text{Candidates: } \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{4} $$

Test candidates using synthetic division

Using the Factor Theorem knowledge point

$$
LATEXBLOCK0
$$

Thus, \(x = 2\) is a rational root. Dividing \(f(x)\) by \(x - 2\):

$$ f(x) = (x - 2)(4x^2 - 5x - 1) $$

Find remaining roots

Using the Quadratic Formula knowledge point

$$
LATEXBLOCK1
$$

Since \(41\) is not a perfect square, these two roots are irrational.

Count rational roots

The roots of \(f(x)\) are:

$$ x_1 = 2 \quad (\text{rational}), \quad x_{2,3} = \frac{5 \pm \sqrt{41}}{8} \quad (\text{irrational}) $$

Thus, exactly \(1\) root is a rational number.

Answer:

• 0
• 1 (Correct answer)
• 2
• 3