


Find the polynomial equation given the following roots.

1.     -2, 1, 4
2.     -1, 1, 3, 5

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:
\[
f(x) = x^4 - 8x^3 + 14x^2 + 8x - 15
\]

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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.

Find the polynomial equation given the following roots.

1. -2, 1, 4
2. -1, 1, 3, 5

Learn how to find polynomial equations from given roots. This step-by-step guide walks you through solving polynomials with roots like -2, 1, 4 and -1, 1, 3, 5, covering polynomial expansion and the Fundamental Theorem of Algebra. Suitable for students in Grades 9-12.

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