Problem 15 - College Algebra

Question

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

Step-by-Step Solution

Answer

The roots of the equation \(x^3 - 2x^2 - 11x + 12 = 0\) are \(x = 1, x = 4, x = -3\).

Step 1: List possible rational roots

Use the Rational Root Theorem to list all possible rational roots. The possible rational roots are \(\pm\) factors of the constant term (12 in this case) divided by the factors of the leading coefficient (1 in this case). So the possible rational roots are: \(\pm1, \pm2, \pm3, \pm4, \pm6, \pm12\).

Step 2: Synthetic division to find a root

Use synthetic division to test the possible rational roots. Start with 1. Set up a synthetic division table: on top, write the coefficients of the cubic polynomial (\(1, -2, -11, 12\)), with 1 to the left. Draw lines under the coefficients and add down columns: Write 1, then \((1)(1) = 1\) and \(-2 + 1 = -1\). Then, \((-1)(1) = -1\) and \(-11 - 1 = -12\). Lastly, \((-12)(1) = -12\) and \(12 - 12 = 0\). The 0 remainder shows that \(x = 1\) is a root of the equation.

Step 3: Solve polynomial

The coefficients from synthetic division are the coefficients of the depressed equation, \(x^2 - x - 12 = 0\). Factor the quadratic to solve for the other roots. The factors of the quadratic equation are \((x - 4)(x + 3) = 0\). Therefore, the solutions to the equation are \(x = 1, x = 4, x = -3\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool in algebra for finding the potential rational roots of a polynomial equation. A rational root can be written as a fraction where both the numerator and the denominator are integers. According to the theorem, if a polynomial has a rational root \frac{p}{q}, then p is a factor of the constant term and q is a factor of the leading coefficient.

For example, for the given polynomial equation \( x^{3}-2 x^{2}-11 x+12=0 \), the constant term is 12 and the leading coefficient is 1. Applying the Rational Root Theorem, this gives us a set of possible rational roots as \( \pm1, \pm2, \pm3, \pm4, \pm6, \pm12 \), simplifying our search for the actual roots of the equation significantly. This theorem can be especially helpful before using synthetic division to test the probable roots systematically.

Polynomial Roots

Finding the roots of a polynomial is a fundamental aspect of solving polynomial equations, as these roots represent the values for which the polynomial evaluates to zero. For a polynomial of degree n, there can be up to n real roots. This includes both the rational roots we can find using the Rational Root Theorem, and other types of roots, such as irrational or complex numbers.

In our example, the polynomial \( x^{3}-2 x^{2}-11 x+12=0 \) is a cubic equation, indicating the possibility of up to three real roots. After testing possible rational roots, synthetic division confirmed that 1 is a root. The remaining roots are determined after factoring the depressed equation resulting from the division process. Understanding how to uncover these roots is essential for solving the entirety of polynomial equations.

Factoring Polynomials

Factoring polynomials involves breaking them down into simpler, multiplicative components or 'factors' that when multiplied together, result in the original polynomial. This process can simplify solving polynomial equations considerably.

After determining one root using synthetic division, we're often left with a lower-degree polynomial, also known as the depressed polynomial. In the provided exercise, we get \(x^2 - x - 12 = 0\) after extracting one root. This is a quadratic equation which we can factor further. Factoring it, we find \(x - 4\) and \(x + 3\) are its factors, which gives us the remaining roots of the polynomial, \(x = 4\) and \(x = -3\). Factoring is crucial in finding all solutions to a polynomial equation and demonstrating that we've found all possible roots, especially when synthetic division can no longer be applied.