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 $-\frac{b}{a}$.