To find the polynomial equation given the roots, we use the fact that if $r_1, r_2, r_3, \dots$ are the roots of the polynomial, then the polynomial can be written as:

$f(x) = (x - r_1)(x - r_2)(x - r_3) \dots$

1. Roots: -2, 1, 4

The polynomial will be:

$f(x) = (x - (-2))(x - 1)(x - 4) = (x + 2)(x - 1)(x - 4)$

Now, let's expand the factors:

$f(x) = (x + 2)(x^2 - 5x + 4)$

Expanding this:

$f(x) = x(x^2 - 5x + 4) + 2(x^2 - 5x + 4)$ $f(x) = x^3 - 5x^2 + 4x + 2x^2 - 10x + 8$ $f(x) = x^3 - 3x^2 - 6x + 8$

Thus, the polynomial is: $f(x) = x^3 - 3x^2 - 6x + 8$

2. Roots: -1, 1, 3, 5

The polynomial will be:

$f(x) = (x - (-1))(x - 1)(x - 3)(x - 5) = (x + 1)(x - 1)(x - 3)(x - 5)$

First, let's simplify the first two factors:

$(x + 1)(x - 1) = x^2 - 1$

Now, multiply this with the remaining factors:

$f(x) = (x^2 - 1)(x - 3)(x - 5)$

Expand the first two factors:

$(x^2 - 1)(x - 3) = x^3 - 3x^2 - x + 3$

Now, multiply this with $(x - 5)$:

$f(x) = (x^3 - 3x^2 - x + 3)(x - 5)$

Expanding:

$f(x) = x^4 - 5x^3 - 3x^3 + 15x^2 - x^2 + 5x + 3x - 15$ $f(x) = x^4 - 8x^3 + 14x^2 + 8x - 15$

Thus, the polynomial is: $f(x) = x^4 - 8x^3 + 14x^2 + 8x - 15$

---

Would you like further details or have any questions?

Relative Questions:

1. How do we determine the degree of a polynomial from its roots?
2. What happens if a root is repeated in the polynomial?
3. How do you factor a cubic polynomial?
4. What methods can we use to solve higher-degree polynomials?
5. How does the Fundamental Theorem of Algebra relate to the number of roots?

Tip: When expanding polynomials, always combine like terms carefully to avoid mistakes in the final expression.