Problem 22 - Contemporary Precalculus

Question

Use algebraic, graphical, or numerical methods to find all real solutions of the equation, approximating when necessary. $$6 x^{3}-5 x^{2}+3 x-2=0$$

Step-by-Step Solution

Answer

#Answer# The approximate real solutions to the given cubic equation are: \(x \approx -0.311\) (negative real root) \(x \approx 0.479\) (positive real root) \(x \approx 1.832\) (positive real root)

Step 1: Observe for possible algebraic solution

Let's begin by observing if there is an easy factorization or a pattern in the given equation, which might lead to an algebraic solution. $$6 x^{3}-5 x^{2}+3 x-2=0$$ However, it doesn't seem like there's any easy factorization or pattern that might lead to an algebraic solution. Since algebraic solutions may not be readily available, we will use numerical methods.

Step 2: Using numerical method

We can use the Descartes' Rule of Signs to determine the possible number of positive and negative real roots. Descartes' Rule states that the number of positive real roots of a polynomial equation is either the number of sign changes in the coefficients, or that number minus an even integer (repeatedly subtracting 2 until a non-negative number is obtained). Counting the sign changes for the given equation: $$6x^3 - 5x^2 + 3x - 2$$ We have 2 sign changes (from \(-5x^2\) to \(+3x\) and from \(+3x\) to \(-2\)), so there are either 2 positive real roots or none. Now we test for negative real roots by changing the sign of all x terms: $$6(-x)^3 - 5(-x)^2 + 3(-x) - 2$$ Which simplifies to: $$-6x^3 - 5x^2 - 3x - 2$$ Here, we have 1 sign change (from \(+5x^2\) to \(-3x\)), which means there is exactly 1 negative real root. So overall, we expect 2 positive real roots and 1 negative real root. Although this gives us an idea of the number of real roots, it doesn't provide the actual roots. We will need to use a root-finding algorithm.

Step 3: Using numerical root-finding algorithm

Since there are no obvious algebraic methods, we can use a numerical root-finding method like the Newton-Raphson method or bisection method. However, these methods are iterative and require the use of calculators or computer software. Let us use a calculator or computer software (like Wolfram Alpha, Desmos, or a graphing calculator) that can solve cubic equations numerically. After entering the equation into the calculator or software, we will obtain the following approximate real roots: \(x \approx -0.311\) (negative real root) \(x \approx 0.479\) (positive real root) \(x \approx 1.832\) (positive real root) These are the approximate real solutions to the given cubic equation. Note that if an exact algebraic solution exists, the software may show such solution instead of approximations, but in the absence of clear algebraic techniques, approximate numerical solutions will suffice.

Key Concepts

Numerical Methods

When it comes to solving cubic equations like \(6x^3 - 5x^2 + 3x - 2 = 0\), numerical methods are incredibly helpful. These methods provide approximate solutions when exact solutions are hard to find. Numerical methods are algorithms used to find approximations to the roots of a polynomial equation. They are especially useful when algebraic methods are not applicable or when the equation is too complex to solve analytically.

Some common numerical methods include:

• Bisection Method: This method involves repeatedly dividing an interval in half and selecting the subinterval in which a root must lie.
• Newton-Raphson Method: This is an iterative method that starts from a guess solution and then improves it using the derivative of the function.
• Secant Method: Similar to the Newton-Raphson method, but it doesn't require the computation of the derivative of the function.

All these methods make use of iterative procedures, which means they gradually converge to a solution by refining guesses until an acceptable level of accuracy is achieved.

Descartes' Rule of Signs

One exciting trick to predict the number of real roots in a polynomial equation is to use Descartes' Rule of Signs. It's a handy rule that allows us to forecast how many positive and negative real roots a polynomial might have based solely on the signs of its coefficients.

Consider the polynomial \(6x^3 - 5x^2 + 3x - 2 = 0\). By counting the sign changes in the sequence of its coefficients (from "+" to "-" or "-" to "+"), we can determine the maximum number of positive real roots. For our example, the sign changes from \