The polynomial of degree 3 , P ( x ) , has a root of multiplicity 2 at x = 1 and a root of multiplicity 1 at x = − 1 . The y -intercept is y = − 0.1 .

Question

The polynomial of degree , , has a root of multiplicity  at  and a root of multiplicity  at . The -intercept is .

Find a formula for P(x):



Answer 1

P(x) = -0.1 * (x-1)(x-1)(x+1)

Answer 2

"Root of multiplicity 2 at x = 1" means that P(x) = 0 when x = 1, and thus P(x) has (x - 1) as one of its factors. The multiplicity 2 indicates that this factor is responsible for nominally 2 roots. In short, (x - 1)² is a factor of P(x).

Likewise, "root of multiplicity 1 at x = -1" means that P(x) = 0 when x = -1, so (x + 1) is also one of the factors.

From this, we can say that P(x) looks like a(x - 1)²(x+1) where a is a constant. This is where we use the y-intercept of -0.1. The y-intercept is the value of y when x = 0, so we should get P(0) = a*(-1)²&1 = a = -0.1

Now that we figured out the constant a, P(x) = -0.1(x-1)²(x+1).

If you'd like the expanded version, it will be P(x) = -0.1x³+0.1x²+0.1x-0.1.