Question

Find a third degree polynomial function of the lowest degree that has the zeros below and whose leading coefficient is one.

-1,0,6

Answer 1

The polynomial function of the lowest degree that has zeroes at -1, 0 and 6 and with a leading coefficient of one is
`p(x) = x^(3)-5\cdot x^(2)-6\cdot x`.

Explanation:

From Fundamental Theorem of Algebra, we remember that the degree of the polynomials determine the number of roots within. Since we know three roots, then the factorized form of the polynomial function with the lowest degree is:

`p(x) = (x-r_(1))\cdot (x-r_(2))\cdot (x-r_(3)) = 0` (1)

Where
`r_(1)`,
`r_(2)` and
`r_(3)` are the roots of the polynomial.

If we know that
`r_(1) = -1`,
`r_(2) = 0` and
`r_(3) = 6`, then the polynomial function in factorized form is:

`p(x) = (x+1)\cdot x \cdot (x-6)` (2)

And by Algebra we get the standard form of the function:

`p(x) = x\cdot (x+1)\cdot (x-6)`

`p(x) = x\cdot (x^(2)-5\cdot x -6)`

`p(x) = x^(3)-5\cdot x^(2)-6\cdot x` (3)

The polynomial function of the lowest degree that has zeroes at -1, 0 and 6 and with a leading coefficient of one is
`p(x) = x^(3)-5\cdot x^(2)-6\cdot x`.