Introduction

This article provides a detailed exploration of finding the zeros of the polynomial function (x^3 - 2x^2 - 5x 6 0). We will employ the Factor Theorem, the Rational Roots Theorem, and basic algebraic techniques to determine the roots of this polynomial.

Using the Factor Theorem

The Factor Theorem states that if (x - c) is a factor of a polynomial (P(x)), then (P(c) 0). Let's apply this theorem to our given polynomial (P(x) x^3 - 2x^2 - 5x 6).

Step 1: Inspect and Test Possible Roots

By inspection, we notice that (x 1) is a root of the polynomial:

(1^3 - 2(1^2) - 5(1) 6 1 - 2 - 5 6 0).

Therefore, (x - 1) is a factor. We can use polynomial division to factor (P(x)) further.

Step 2: Polynomial Division

Performing polynomial division of (P(x)) by (x - 1), we get:

[frac{x^3 - 2x^2 - 5x 6}{x - 1} x^2 - x - 6]

Step 3: Factor the Quotient

The quotient (x^2 - x - 6) can be factored as:

[(x - 3)(x 2)]

Thus, the polynomial (P(x)) can be written as:

[(x - 1)(x - 3)(x 2)]

Step 4: Determine the Zeros

The zeros of (P(x)) are the roots of the equation:

[(x - 1)(x - 3)(x 2) 0]

Setting each factor to zero, we get:

[begin{align*}x - 1 0 quad Rightarrow quad x 1 x - 3 0 quad Rightarrow quad x 3 x 2 0 quad Rightarrow quad x -2end{align*}]

Advanced Methods

Another approach to finding the zeros of the polynomial (P(x) x^3 - 2x^2 - 5x 6) involves the Rational Roots Theorem, which is based on the assumption that the rational roots, if any, must be of the form (frac{p}{q}), where (p) is a factor of the constant term and (q) is a factor of the leading coefficient.

Step 1: Identify Possible Rational Roots

The constant term is 6, and its factors are (pm 1, pm 2, pm 3, pm 6). The leading coefficient is 1, and its factors are (pm 1). Therefore, the possible rational roots are (pm 1, pm 2, pm 3, pm 6).

Step 2: Test the Possible Roots

Substituting these possible roots into (P(x)):

[begin{align*}P(1) 1^3 - 2(1^2) - 5(1) 6 0 P(-1) (-1)^3 - 2(-1)^2 - 5(-1) 6 0 1 5 6 8 quad text{(not a zero)} P(2) 2^3 - 2(2^2) - 5(2) 6 8 - 8 - 10 6 -4 quad text{(not a zero)} P(-2) (-2)^3 - 2(-2)^2 - 5(-2) 6 -8 - 8 10 6 0 P(3) 3^3 - 2(3^2) - 5(3) 6 27 - 18 - 15 6 0end{align*}]

We find that (x 1), (x -2), and (x 3) are the roots of the polynomial.

Conclusion

The zeros of the polynomial function (x^3 - 2x^2 - 5x 6 0) are (x 1), (x -2), and (x 3). These roots can be found using the Factor Theorem, polynomial division, and the Rational Roots Theorem.

This method is particularly useful in solving higher degree polynomial equations with rational coefficients. By employing these techniques, we can efficiently determine the roots of a polynomial and understand its behavior.