write a polynomial of degree 3 whose only roots are x=2 and x =2/5


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.

write a polynomial of degree 3 whose only roots are x=2 and x =2/5

Discover how to construct a polynomial of degree 3 with the roots x = 2 and x = 2/5. This solution explains factorization, root multiplicity, and expanding binomials. Suitable for high school students, particularly in Grades 9-12.

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