What is a cubic polynomial function in standard form with zeros 3, -4, and 5?

2 Answers
Feb 16, 2018

y=x^3-4x^2-17x+60

Explanation:

.

The standard form of a cubic polynomial function is:

y=ax^3+bx^2+cx+d

If we know the zeros as being alpha, beta, and theta then we know its factored form to be:

y=(x-alpha)(x-beta)(x-theta)

simply because you would have had to set the expression within each set of parentheses equal to 0 and solve for a root.

Therefore, we can write the polynomial in this problem as:

y=(x-3)(x+4)(x-5)

Now, if we multiply these through and remove the parentheses we will have the standard form of it:

y=(x-3)(x^2-x-20)

y=x^3-x^2-20x-3x^2+3x+60

y=x^3-4x^2-17x+60

Feb 16, 2018

x^3-4x^2-17x+60

Explanation:

There is only one cubic polynomial in standard form (i.e. monic) with three given distinct zeros, a, b and c.
It is given by the product:

(x-a)(x-b)(x-c)

In your case, it is

(x-3)(x+4)(x-5)=(x^2+x-12)(x-5)
=x^3-4x^2-17x+60