To create a polynomial with 4 roots, let's follow this general rule:

If a polynomial has roots \( r_1, r_2, r_3, r_4 \), then the polynomial can be written as:

\[
P(x) = a(x - r_1)(x - r_2)(x - r_3)(x - r_4)
\]

Here, \( a \) is any non-zero constant (it affects the scaling of the polynomial, but not the roots).

### Example:
Suppose we choose the roots as \( r_1 = 1 \), \( r_2 = -2 \), \( r_3 = 3 \), and \( r_4 = 4 \).

Then the polynomial will be:

\[
P(x) = a(x - 1)(x + 2)(x - 3)(x - 4)
\]

For simplicity, let's set \( a = 1 \):

\[
P(x) = (x - 1)(x + 2)(x - 3)(x - 4)
\]

Now, let's expand this expression:

\[
P(x) = [(x - 1)(x + 2)] \cdot [(x - 3)(x - 4)]
\]

First, expand the two binomials:

\[
(x - 1)(x + 2) = x^2 + x - 2
\]
\[
(x - 3)(x - 4) = x^2 - 7x + 12
\]

Now multiply the two quadratic expressions:

\[
P(x) = (x^2 + x - 2)(x^2 - 7x + 12)
\]

Expanding this:

\[
P(x) = x^4 - 7x^3 + 12x^2 + x^3 - 7x^2 + 12x - 2x^2 + 14x - 24
\]

Simplify the terms:

\[
P(x) = x^4 - 6x^3 + 3x^2 + 26x - 24
\]

So the polynomial with roots 1, -2, 3, and 4 is:

\[
P(x) = x^4 - 6x^3 + 3x^2 + 26x - 24
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do we determine the degree of a polynomial based on its roots?
2. Can a polynomial have complex roots?
3. What happens if the coefficient \( a \) is not equal to 1?
4. How can we factor a polynomial if the roots are known?
5. What is the connection between the number of roots and the Fundamental Theorem of Algebra?

**Tip:** The sum of the roots of a polynomial \( ax^n + bx^{n-1} + \cdots \) is \( -\frac{b}{a} \).

Discover how to construct a polynomial with specified roots. This example uses the roots 1, -2, 3, and 4 to demonstrate the process, expanding and simplifying the polynomial step-by-step. Suitable for high school students in Grades 10-12.

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