Question

The polynomial of degree 4, P(x) has a root of multiplicity 2 at x=1 and roots of multiplicity 1 at x=0 and x=-3. It goes through the point (6,8100). Find the formula for P(x)

Answer 1

The formula for the polynomial P(x) of degree 4 with given roots and point is P(x) = 6(x)(x - 1)^2(x + 3).

Step-by-step explanation:

To find the formula for the polynomial P(x) of degree 4 which has roots of multiplicity 2 at x=1, and roots of multiplicity 1 at x=0 and x=-3, and passes through the point (6, 8100), we start by constructing the polynomial's factored form.

Given the roots and their multiplicities, the polynomial can be expressed as:

P(x) = a(x - 0)(x - 1)^2(x + 3),

where a is a leading coefficient that we need to determine. Utilizing the point (6, 8100) which the polynomial passes through, we have:

8100 = a(6 - 0)(6 - 1)^2(6 + 3),

8100 = a(6)(5)^2(9),

8100 = 1350a,

We find the value of a by dividing both sides by 1350:

a = 8100 / 1350,

a = 6.

Now we can write the full polynomial as:

P(x) = 6(x)(x - 1)^2(x + 3).