91Ó°ÊÓ

Understanding polynomial roots is a fundamental concept in algebra, especially when solving polynomial equations. Simply put, a root of a polynomial is a value of the variable that makes the polynomial equal to zero. In our exercise, we are concerned with the polynomial \( \frac{2}{3} x^{3} - x^{2} - 5x + 2 = 0 \). By finding its roots, we determine the \( x \)-values at which the polynomial crosses or touches the \( x \)-axis.

It's important to note that polynomials can have different types of roots:

• Real roots: These are roots that are real numbers and can be rational or irrational.
• Complex roots: These are roots that involve imaginary numbers and appear in conjugate pairs when coefficients are real.

For this exercise, we're specifically looking for rational roots, which are possibly expressible as simple fractions.

Rational solutions, also known as rational roots, are solutions that can be expressed as a quotient of two integers. When using the Rational Root Theorem to find such solutions for a polynomial, we're essentially narrowing down potential candidates that can be checked. The theorem states that if a polynomial has a rational root \( \frac{p}{q} \), then:

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

In our exercise, the polynomial \( 2x^{3} - 3x^{2} - 15x + 6 = 0 \) was derived after eliminating fractions for simplicity. Here, we observed that the constant term is 6 and the leading coefficient is 2.

After finding their factors (\( \pm 1, \pm 2, \pm 3, \pm 6 \) for 6, and \( \pm 1, \pm 2 \) for 2), we can form possible rational roots: \( \pm 1, \pm 2, \pm 3, \pm 6, \pm \frac{1}{2}, \pm \frac{3}{2} \). These are potential candidates to be verified using substitution or other methods like synthetic division.

Factor analysis in the context of solving polynomial equations is a method to simplify the finding of roots. By identifying factors, we can break down complex expressions into simpler products—much like simplifying fractions in arithmetic.

The first crucial step in our exercise was performing factor analysis on the polynomial by eliminating fractions. This resulted in a cleaner expression \( 2x^{3} - 3x^{2} - 15x + 6 = 0 \). With fractions removed, identifying factors becomes easier.

Next comes finding the factors of the constant and the leading coefficient:

• Constant term (6): Factors are \( \pm 1, \pm 2, \pm 3, \pm 6 \).
• Leading coefficient (2): Factors are \( \pm 1, \pm 2 \).

Once these are known, they guide the formation of potential rational roots by pairing all factors of the constant term with all factors of the leading coefficient. This systematic approach reduces guesswork and efficiently highlights possible solutions for thorough verification.