Question

Write a polynomial function of least degree with integral coefficients that has the given zeros. ​

Answer 1

The roots of the polynomial are provided as 2, -3, and 4.

A polynomial with roots a, b, and c takes the structure of:

p(x) = (x - a)(x - b)(x - c)

Let's substitute the roots into this structure:

p(x) = (x - 2)(x + 3)(x - 4)

To find the polynomial function, we need to expand and simplify this equation. Let's start out by multiplying the first two factors:

p(x) = (x² - 2x + 3x - 6)(x - 4)
= (x² + x - 6)(x - 4)

Next, we will distribute the (x - 4) term:

p(x) = x³ + x² - 4x² - 6x + 4x - 24
= x³ - 3x² - 2x - 24

Combine like terms to simplify:

p(x) = x³ - 3x² - 8x + 24

So, the polynomial function of least degree with integral coefficients that has the given zeros (2, -3, and 4) is:

p(x) = x³ - 3x² - 8x + 24

Answer 2

Explanation:

To get the equation, you must create the factors containing the roots.

So for example the first set of roots can be produced from

(x -2)(x^2 + 9)

which equals (when expanded) x^3 + 9x - 2x^2 - 18 which when put in the correct order gives

x^3 -2x^2 + 9x - 18

The second one is a bit more difficult to judge.

-5 and -3+ square root(2)

I don't think this is a polynomial. The coefficent is not integral. (-3 +√2)