91Ó°ÊÓ

The Rational Root Theorem helps us find potential rational zeros of a polynomial. It states that any rational zero, written as \(\frac{p}{q}\), must be a factor of the constant term (\(p\)) divided by a factor of the leading coefficient (\(q\)).
This means two things:
1. Identify the factors of the constant term.
2. Identify the factors of the leading coefficient.
For the polynomial \(M(t) = 18t^3 - 21t^2 + 10t - 2\), the constant term is \(-2\) and the leading coefficient is \(18\).
The possible factors of \(-2\) are \(\pm 1, \pm 2\). The possible factors of \(18\) are \(\pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18\).
This results in possible rational zeros: \(\pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{1}{9}, \pm \frac{1}{18}\).
You can test these values using synthetic division to check if they yield a remainder of zero and thus confirm if they are exact zeros.

Synthetic division is a simplified form of polynomial division. It helps in testing potential zeros of polynomials quickly. To perform synthetic division, follow these steps:
1. Write down the coefficients of the polynomial.
2. Choose a possible zero to test.
3. Perform synthetic division steps:

• Bring down the first coefficient.
• Multiply the possible zero by this first coefficient and add it to the next coefficient.
• Repeat the process with the resulting sum and move through all coefficients.

If the final sum (remainder) is zero, the tested value is a zero of the polynomial.
For example, with the polynomial \(M(t) = 18t^3 - 21t^2 + 10t - 2\) and testing \(t = \frac{1}{3}\) as potential zero, synthetic division confirms \(t = \frac{1}{3}\) as a zero because the remainder is 0. This means \(t - \frac{1}{3}\) is a factor.

The quadratic formula is used to find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a\), \(b\), and \(c\) are coefficients of the equation.
Steps to use the quadratic formula:
1. Identify the coefficients \(a\), \(b\), and \(c\).
2. Calculate the discriminant \(b^2 - 4ac\).
3. Substitute \(a\), \(b\), and the discriminant into the formula.
For the polynomial \(54t^2 - 3t + 1 = 0\):

• \(a = 54\), \(b = -3\), and \(c = 1\).
• Calculate the discriminant: \((-3)^2 - 4\cdot54\cdot1 = -207\).
• Substitute into the formula: \(t = \frac{3 \pm i\sqrt{207}}{108}\).
• The solutions are complex roots due to the negative discriminant.

Complex roots arise when solving polynomial equations with a negative discriminant. This is indicated by the presence of the imaginary unit \(i\), where \(i = \sqrt{-1}\).
For example, for the quadratic \(54t^2 - 3t + 1 = 0\), the discriminant is \(-207\), which is negative.
Using the quadratic formula, we substitute the discriminant:
\[ t = \frac{3 \pm i \sqrt{207}}{108} \]
This gives two complex roots:

• \(t = \frac{1}{36} + \frac{i \sqrt{207}}{108}\)
• \(t = \frac{1}{36} - \frac{i \sqrt{207}}{108}\)

Therefore, the polynomial \(M(t) = 18t^3 - 21t^2 + 10t - 2\) has one real root \(t = \frac{1}{3}\) and two complex roots.