Math Snap

STEP 1

Assumptions
1. The roots of the polynomial are $-2, 3, 5$.
2. The polynomial is a cubic polynomial because there are three roots.
3. The polynomial has real coefficients.

STEP 2

To find a polynomial with given roots, we use the fact that if $r$ is a root of the polynomial, then $(x - r)$ is a factor of the polynomial.

STEP 3

Write down the factors corresponding to each root.
$$(x - (-2)), (x - 3), (x - 5)$$

STEP 4

Simplify the factors to make them ready for multiplication.
$$(x + 2), (x - 3), (x - 5)$$

STEP 5

Multiply the factors together to get the polynomial.
$$(x + 2)(x - 3)(x - 5)$$

STEP 6

First, multiply the first two factors $(x + 2)(x - 3)$.
$$(x + 2)(x - 3) = x^2 - 3x + 2x - 6$$

STEP 7

Combine like terms from the multiplication.
$$x^2 - 3x + 2x - 6 = x^2 - x - 6$$

STEP 8

Now, multiply the result by the third factor $(x - 5)$.
$$(x^2 - x - 6)(x - 5)$$

STEP 9

Use the distributive property (also known as the FOIL method for binomials) to multiply the terms.
$$x^3 - 5x^2 - x^2 + 5x - 6x + 30$$

STEP 10

Combine like terms.
$$x^3 - 6x^2 - x + 30$$

The polynomial equation with roots of $-2, 3, 5$ is:
$$x^3 - 6x^2 - x + 30 = 0$$ This is the required polynomial equation.