Question Find a polynomial equation with roots at $-2, 3, 5$ .

Studdy Solution

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.
$x^3 - 6x^2 - x + 30$

STEP 11

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

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms