To write a polynomial of degree 3 with the only roots $x = 2$ and $x = \frac{2}{5}$, we can use the fact that each root corresponds to a factor of the polynomial in the form $(x - r)$, where $r$ is a root.

Since $x = 2$ is a root with multiplicity 2 (to ensure the polynomial is degree 3), and $x = \frac{2}{5}$ is a root with multiplicity 1, the polynomial will have the following factors:

$(x - 2)^2 \quad \text{and} \quad \left(x - \frac{2}{5}\right)$

Now, multiply the factors:

$P(x) = (x - 2)^2 \left(x - \frac{2}{5}\right)$

First, expand $(x - 2)^2$:

$(x - 2)^2 = (x - 2)(x - 2) = x^2 - 4x + 4$

Now multiply this by $\left(x - \frac{2}{5}\right)$:

$P(x) = (x^2 - 4x + 4)\left(x - \frac{2}{5}\right)$

Distribute $(x - \frac{2}{5})$ across $x^2 - 4x + 4$:

$P(x) = x(x^2 - 4x + 4) - \frac{2}{5}(x^2 - 4x + 4)$

Simplify each term:

$P(x) = x^3 - 4x^2 + 4x - \frac{2}{5}(x^2 - 4x + 4)$

Distribute $-\frac{2}{5}$:

$P(x) = x^3 - 4x^2 + 4x - \frac{2}{5}x^2 + \frac{8}{5}x - \frac{8}{5}$

Now, combine like terms:

$P(x) = x^3 - \left(4 + \frac{2}{5}\right)x^2 + \left(4 + \frac{8}{5}\right)x - \frac{8}{5}$

$P(x) = x^3 - \frac{22}{5}x^2 + \frac{28}{5}x - \frac{8}{5}$

Finally, multiply the entire polynomial by 5 to eliminate the fractions:

$P(x) = 5x^3 - 22x^2 + 28x - 8$

Thus, the polynomial is:

$P(x) = 5x^3 - 22x^2 + 28x - 8$

Would you like any details on this? Here are some related questions:

1. How do you find a polynomial given different sets of roots?
2. What is the significance of root multiplicity in polynomials?
3. How do you expand products of binomials like $(x - a)^2$?
4. Can a polynomial of degree 3 have more than two distinct roots?
5. What happens to the graph of a polynomial when a root has multiplicity greater than 1?

Tip: Roots with multiplicity greater than 1 flatten the graph at those points, creating a "bounce" effect at the root.