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To construct a polynomial from the given roots, we follow these steps:

The roots are \( \frac{-2}{5} \) and \( \frac{3}{4} \). According to the polynomial root theorem, if \( r \) is a root of the polynomial, then \( (x - r) \) is a factor of the polynomial.

### Step 1: Express the factors
- The factor for the root \( \frac{-2}{5} \) is \( \left( x - \frac{-2}{5} \right) = \left( x + \frac{2}{5} \right) \).
- The factor for the root \( \frac{3}{4} \) is \( \left( x - \frac{3}{4} \right) \).

### Step 2: Eliminate the fractions
We can eliminate the fractions by multiplying each factor by the denominator:
- Multiply \( \left( x + \frac{2}{5} \right) \) by 5 to get \( (5x + 2) \).
- Multiply \( \left( x - \frac{3}{4} \right) \) by 4 to get \( (4x - 3) \).

### Step 3: Multiply the factors
Now, the polynomial is formed by multiplying these two factors:
\[
f(x) = (5x + 2)(4x - 3)
\]

### Step 4: Expand the expression
Expand the product:
\[
f(x) = 5x(4x) + 5x(-3) + 2(4x) + 2(-3)
\]
\[
f(x) = 20x^2 - 15x + 8x - 6
\]
\[
f(x) = 20x^2 - 7x - 6
\]

Thus, the polynomial is:
\[
f(x) = 20x^2 - 7x - 6
\]

Let me know if you would like further clarification!

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### Related Questions:
1. How do you determine the polynomial degree based on the number of roots?
2. What happens if a root is a repeated value (a double root)?
3. How can you factor a polynomial if the roots are complex?
4. How do rational roots affect the behavior of the polynomial graph?
5. How can you verify the roots of a polynomial using synthetic division?

### Tip:
Always simplify polynomial factors by eliminating fractions early on, as it makes expansion easier.

Write the polynomial function with the given roots -2/5 and 3/4.

Learn how to construct a polynomial given the roots -2/5 and 3/4. This example demonstrates factoring, eliminating fractions, and expanding to form the quadratic equation f(x) = 20x^2 - 7x - 6. Suitable for students in Grades 9-12.

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