Category Archives: Sullivan Chapter 5

Form a Polynomial given the Degree and Zeros

Example: Form a polynomial f(x) with real coefficients having the given degree and zeros.

Degree 4; Zeros -2-3i; 5 multiplicity 2

Solution:

By the Fundamental Theorem of Algebra, since the degree of the polynomial is 4 the polynomial has 4 zeros if you count multiplicity.

There are three given zeros of -2-3i, 5, 5.

The remaining zero can be found using the Conjugate Pairs Theorem.  f(x) is a polynomial with real coefficients.  Since -2-3i is a complex zero of f(x) the conjugate pair of -2+3i is also a zero of f(x).

Now that all the zeros of f(x) are known the polynomial can be formed with the factors that are associated with each zero.

Since f(x) has a zero of 5, f(x) has a factor of x-5

Since f(x) has a second zero of 5, f(x) has a second factor of x-5

Since f(x) has a factor of -2-3i, f(x) has a factor of x-(-2-3i)

Since f(x) has a factor of -2+3i, f(x) has a factor of x-(-2+3i)

Form the polynomial using all of the factors.  The leading coefficient will remain unknown.
f(x)=a(x-5)(x-5)(x-(-2-3i))(x-(-2+3i))
Multiply the factors with complex numbers.  Doing so will cancel the complex numbers from the expression

  • Distribute the minus
  • Multiply each term in one factor by each term in the other factor
  • simplify i^2=-1
  • combine like terms
f(x)=a(x-5)(x-5)(x-(-2-3i))(x-(-2+3i))
f(x)=a(x-5)^2(x+2+3i)(x+2-3i)
f(x)=a(x-5)^2(x^2+2x-3ix+2x+4-6i+3ix+6i-9i^2)
f(x)=a(x-5)^2(x^2+2x-3ix+2x+4-6i+3ix+6i-9(-1))
f(x)=a(x-5)^2(x^2+2x-3ix+2x+4-6i+3ix+6i+9)
f(x)=a(x-5)^2(x^2+4x+13)
Multiply the other pair of factors
f(x)=a(x-5)^2(x^2+4x+13)
f(x)=a(x-5)(x-5)(x^2+4x+13)
f(x)=a(x^2-5x-5x+25)(x^2+4x+13)
f(x)=a(x^2-10x+25)(x^2+4x+13)
Multiply the two trinomials by multiplying each term in the first trinomial by each term in the other trinomial and then combine like terms
f(x)=a(x^2-10x+25)(x^2+4x+13)
f(x)=a(x^4+4x^3+13x^2-10x^3-40x^2-130x+25x^2+100x+325)
f(x)=a(x^4-6x^3-2x^2-30x+325)

The polynomial with degree 4 and zeros of -2-3i and 5 wiht multiplicity 2 is f(x)=a(x^4-6x^3-2x^2-30x+325)