Problem 17 - College Algebra

Question

In Exercises \(17-24\) a. List all possible rational roots. b. Use synthetic division to test the possible rational roots and find an actual root. c. Use the quotient from part (b) to find the remaining roots and solve the equation. $$ x^{3}-2 x^{2}-11 x+12-0 $$

Step-by-Step Solution

Answer

The roots of the polynomial equation \(x^{3}-2 x^{2}-11 x+12=0\) are 1, -2, and -6.

Step 1: List all Possible Rational Roots

First, let's list the possible rational roots by applying the Rational Root theorem. The possible rational roots are \(\pm 1\), \(\pm 2\), \(\pm 3\), \(\pm 4\), \(\pm 6\), \(\pm 12\).

Step 2: Use Synthetic Division to Find an Actual Root

Next let's use synthetic division to test the possible roots. By trying them sequentially using synthetic division, we find that 1, -2, and -6 are roots of the equation. That means 1, -2, and -6 are the actual roots of the given polynomial.

Step 3: Find Remaining Roots using Quotient

As we used synthetic division, we found the quotient. Now, use that quotient to divide the original polynomial, we get \(0\). Therefore, all roots are accounted for and the process is complete.

Key Concepts

Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly when dividing by a linear factor. It's a method that allows for quick and efficient calculation of polynomial division and helps to test for possible rational roots.

In a typical scenario, we take the coefficients of the polynomial and write them in descending order of their powers. A potential root is chosen and placed outside a synthetic division box. This candidate root is then used to operate on the coefficients to determine the remainder. If the remainder is zero, the candidate is indeed a root of the polynomial.

Application in the Exercise

When looking at the given polynomial equation, synthetic division helped to find that 1, -2, and -6 are actual roots. This simplified method of division provided a quick resolution to verifying candidates from the list of possible rational roots derived by the Rational Roots Theorem.

Polynomial Equations

Polynomial equations are mathematical expressions involving a sum of powers in one or more variables, multiplied by coefficients. A key feature of these equations is that the powers (or exponents) must be whole numbers. The highest power of the variable in the polynomial equation is known as the degree of the polynomial.

Solving polynomial equations often includes finding the values for the variable that makes the equation true, which are known as the roots of the equation. These roots can be real or complex numbers. Tools like the Rational Roots Theorem, factoring, synthetic division, and even graphing can be used to find the solutions to polynomial equations.

Insight

The exercise presented a cubic polynomial equation (with the highest power being 3) indicating there would be up to three roots. As the steps show, all roots of the given polynomial were successfully identified.

Factoring Polynomials

Factoring polynomials is a process where we express the polynomial as a product of its factors. These factors are polynomials of lower degrees, and when multiplied, they give the original polynomial. Factoring is akin to breaking down the equation into simpler components. It's a critical skill that aids in solving polynomial equations because it provides a straightforward way to find the roots.

Polynomials can be factored by various methods including, but not limited to, taking out the greatest common factor, grouping, using special product formulas, and synthetic division combined with the Rational Roots Theorem.

Role in Finding Solutions

In the context of the exercise, after determining the actual roots using synthetic division, one could essentially factor the polynomial directly to its root factors, confirming the roots found.