91Ó°ÊÓ

Polynomial functions feature expressions involving variables raised to whole number powers, typically arranged in order of decreasing exponent size. These are mathematical sentences that describe a vast array of relationships in mathematics and can be of varying degrees depending on the highest power of the variable involved.

In the provided exercise, we deal with a cubic polynomial function: \(f(x) = x^3 - \frac{3}{2}x^2 - \frac{23}{2}x + 6\). This means it is a polynomial equation of degree 3. The degree of a polynomial gives us important information; for a cubic polynomial like this one, up to three real roots can exist, including rational roots. Understanding the structure of polynomial functions is crucial because it dictates how we approach finding solutions or factoring them into simpler expressions.

The Rational Root Theorem is an invaluable tool when trying to find rational zeros of polynomial equations. This theorem provides a method to identify possible rational roots by examining the factors of the constant term and the leading coefficient of the polynomial.

For the cubic polynomial given, \(f(x) = x^{3} - \frac{3}{2}x^{2} - \frac{23}{2}x + 6\), the Rational Root Theorem suggests that any rational root can be expressed as \(\frac{p}{q}\), where \(p\) is a factor of the constant 6, and \(q\) is a factor of the leading coefficient, which is 1 in this case:

• Possible values for \(p\) are: \(\pm 1, \pm 2, \pm 3, \pm 6\)
• Possible value for \(q\) is: \(\pm 1\)

Thus, the potential rational roots for this polynomial equation are \(\pm 1, \pm 2, \pm 3, \pm 6\). By substituting these values into the polynomial, one can determine which actually result in zero, confirming them as rational roots.

Once the rational roots are determined, they can be used to factor the polynomial into simpler components. Factoring polynomials involves expressing the original polynomial as a product of its roots.

In this exercise, the polynomial \(x^{3} - \frac{3}{2}x^{2} - \frac{23}{2}x + 6\) has been tested to confirm rational roots of \(-2, 1.5,\) and \(2\). These roots indicate the polynomial can be factored neatly into linear factors that represent each root:

• A root of \(2\) translates to the factor \((x - 2)\)
• A root of \(-2\) gives the factor \((x + 2)\)
• A root of \(1.5\) can be expressed as \((x - 1.5)\)

Thus, the polynomial can be expressed as the product of these factors, \((x - 2)(x + 2)(x - 1.5)\), allowing us to see the simpler underlying structure of the original polynomial.