Question

Enter the polynomial function with the least degree and a leading coefficient of 1 that has the given zeros. 0,4, square root of 6

Answer 1

The equation of the polynomial function is f(x) = x⁴ - 4x³ - 6x² + 24x

How to determine the equation of the polynomial function

From the question, we have the following parameters that can be used in our computation:

Zeroes: x = 0, 4, √6

All with a multiplicity of 1

Also, we have

Leading coefficient, a = 1

First, we do the following

x = √6

This gives

x² = 6

So, we have

x² - 6 = 0

The other zeros are

x = 0, 4

So, we have

x - 0 = 0 and x - 4 = 0

To get the polynomial function, we multiply the leading coefficient (a), the factors x² - 6, x - 0 and x - 4

So, we have

f(x) = a(x - 0)(x - 4)(x² - 6)

Recall that the leading coefficient (a) is

a = 1

This gives

f(x) = (x - 0)(x - 4)(x² - 6)

Evaluate

f(x) = x(x - 4)(x² - 6)

Expand

f(x) = (x² - 4x)(x² - 6)

So, we have

f(x) = x⁴ - 6x² - 4x³ + 24x

Rearrange the terms

f(x) = x⁴ - 4x³ - 6x² + 24x

Hence, the equation is f(x) = x⁴ - 4x³ - 6x² + 24x

Answer 2

`x^(3)-(4+√(6))x^(2)+4√(6)x`

Explanation:

The roots given are 0,4 and
`√(6)`.

We are required to find the polynomial function with the least degree

having the roots as 0,4 and
`√(6)`.The minimum number of roots it can have is three and hence the least degree polynomial should be a cubic function.Hence the polynomial is of the form:
`f(x)=ax^(3)+bx^(2)+cx+d`

where a=1 (given in the question)

sum of the roots =
`-(b)/(a)`=
`4+√(6)`

∴b=
`-(4+√(6))`

product of the roots taken two at a time =
`(c)/(a)=4√(6)`

∴c=4
`√(6)`

product of the roots =
`-(d)/(a)` = 0

∴d=0

Hence the polynomial function becomes
`x^(3)-(4+√(6))x^(2)+4√(6)x`