Question

The polynomial of degree 3 3 , P ( x ) P ( x ) , has a root of multiplicity 2 2 at x = 1 x = 1 and a root of multiplicity 1 1 at x = − 1 x = - 1 . The y y -intercept is y = − 0.8 y = - 0.8 . Find a formula for P ( x ) P ( x ) . P ( x ) =

Answer 1

A formula for P(x) is;

P(x) = -0.8·x³ + 0.8·x² + 0.8·x - 0.8

Explanation:

The given parameters of the polynomial are;

The degree of the polynomial = 3

The multiplicity of the root at x = 1 is 2

The multiplicity of the root at x = -1 is 1

The y-intercept is y = -0.8

Therefore, we have to find the formula for the three degree polynomial;

The factors of the three degree polynomial are (x - 1)², and (x + 1)

Multiplying the factors together, we get;

(x - 1)²×(x + 1) = x³ - x² - x + 1

f(0) = -0.8, therefore, we get;

P(x) = -0.8 × (x - 1)² × (x + 1) = -0.8·x³ + 0.8·x² + 0.8·x - 0.8

The required polynomial is;

P(x) = -0.8·x³ + 0.8·x² + 0.8·x - 0.8