Determine a polynomal of degree 3 in general from with data zeros -1(double) and -4 ; p(-2)=6

Solution:
1. Given zeros and multiplicities:
* Zeros: -1 (double root), -4

2. General form of the polynomial with given zeros:
p(x) = a(x + 1)^2(x + 4)

3. Use given point to find the coefficient a:
* Given: p(-2) = 6
* Substitute x = -2 into the polynomial:
p(-2) = a(-2 + 1)^2(-2 + 4)
6 = a(-1)^2(2)
6 = 2a
a = 3

4. The polynomial in general form:
p(x) = 3(x + 1)^2(x + 4)

5. Expand the polynomial to standard form:
p(x) = 3(x^2 + 2x + 1)(x + 4)
p(x) = 3(x^3 + 4x^2 + 2x^2 + 8x + x + 4)
p(x) = 3(x^3 + 6x^2 + 9x + 4)
p(x) = 3x^3 + 18x^2 + 27x + 12