Find a polynomial of minimum degree with zeros -3, -2 and 1 (of multiplicity 2)
f(x) = (x+3)(x+2)(x-1)^2
To find a polynomial with the given zeros, you need to determine the factors of the polynomial.
From the given zeros, we have -3, -2, and 1 (with multiplicity 2). So, the factors of the polynomial are given by (x+3), (x+2), and (x-1)^2.
To find the polynomial, multiply these factors together:
(x+3)(x+2)(x-1)^2
Expanding this expression, we get:
(x+3)(x+2)(x-1)^2 = (x^2 + 5x + 6)(x-1)^2
Next, we can simplify further by expanding the squared term:
(x^2 + 5x + 6)(x-1)(x-1)
Expanding this, we get:
(x^2 + 5x + 6)(x^2 - 2x + 1)
Multiplying each term, we obtain:
x^4 + 3x^3 - 4x^2 - 4x + 6
So, the polynomial of minimum degree with zeros -3, -2 (each with multiplicity 1), and 1 (with multiplicity 2) is:
f(x) = x^4 + 3x^3 - 4x^2 - 4x + 6