91Ó°ÊÓ

Understanding the degree of a polynomial is crucial in finding its zeros. The degree is the highest power of the variable in the polynomial. In the given example, \[ U(x) = x^{4} - 4x^{3} + x^{2} + 12x - 12, \] the polynomial degree is 4 because the highest power of \(x\) is 4. The degree tells us the maximum number of roots (real and complex) that we can expect. So, for a fourth-degree polynomial, you can find up to 4 roots.

The Rational Root Theorem helps in identifying possible rational roots of a polynomial. According to this theorem, if \(p/q\) is a root of the polynomial, then:

• \(p,\) is a factor of the constant term.
• \(q,\) is a factor of the leading coefficient.

For the polynomial \[ U(x) = x^{4} - 4x^{3} + x^{2} + 12x - 12, \] the constant term is -12, and the leading coefficient is 1. Thus, the possible rational roots are: \[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12. \] By testing each of these, we determine if they are indeed roots of the polynomial.

Synthetic division is a method for dividing a polynomial by a binomial of the form \(x - c\). It simplifies polynomial division significantly compared to long division. Let's see it in action for the polynomial \[ U(x) = x^{4} - 4x^{3} + x^{2} + 12x - 12, \] where we found that 2 is a root. This shows that \(x - 2\) is a factor of the polynomial.
Setting up the synthetic division:

\begin{array}{r|rrrrr} 2 & 1 & -4 & 1 & 12 & -12 \ & & 2 & -4 & -6 & 12 \ \text{} & 1 & -2 & -3 & 6 & 0 \end{array}\end{array} The quotient from the division is another polynomial: \( x^{3} - 2x^{2} - 3x + 6\). We can now repeat the process with this new polynomial to find its roots, continuing until we fully factorize the original polynomial.

Complex roots come in pairs known as conjugate pairs. These occur when solving quadratic factors that do not have real solutions. For example, if we factorize \[ x^{2} + 1, \] we set it to zero to find the roots: \[ x^{2} + 1 = 0 \] Solving this gives us: \[ x = \pm i, \] where \(i\) is the imaginary unit (\(i = \sqrt{-1}\)). In the given polynomial, after performing synthetic division, one of the factors was \[ x^{2} + 1, \] leading us to the complex roots \(i\) and \(-i\). Therefore, the polynomial \[ U(x) = x^{4} - 4x^{3} + x^{2} + 12x - 12, \] has 2 real roots and 2 complex roots.