Finding the function of a polynomial

Question

A degree 4 polynomial with integer coefficients has zeros −5−2i and 1, with 1 a zero of multiplicity 2. If the coefficient of x^4 is 1, what is the function of the polynomial

Answer 1

With a degree 4 polynomial, you need four factors:

 

p(x) = a (x+b)(x+c)(x+d)(x+e)

 

We need to find the values of a, b, c, d, and e.  The coefficient of x⁴ is one, so a = 1.  Two roots are given, -5-2i and 1.  If -5-2i is a root, then so is its conjugate -5+2i; so we have three roots.  The root 1 has a multiplicity of 2 , meaning it is counted as a root twice.  So we have:

• a = 1
• b = -5-2i
• c = -5+2i
• d = 1
• e = 1

 

p(x) = a (x+b)(x+c)(x+d)(x+e)

p(x) = (x+(-5-2i))(x+(-5+2i))(x-1)²

 

Multiply it out to get the polynomial in standard form