Problem 15 - A Graphical Approach to Precalculus with Limits

Question

Find all complex solutions of each equation. Do not use a calculator. $$3 x^{3}+2 x^{2}-3 x-2=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 1\), \(x = -\frac{2}{3}\), and \(x = -1\).

Step 1: Identify Possible Rational Roots

We start by applying the Rational Root Theorem, which suggests that any rational solution is a factor of the constant term divided by a factor of the leading coefficient. For the equation \(3x^3 + 2x^2 - 3x - 2 = 0\), the constant term is \(-2\) and the leading coefficient is \(3\). Therefore, the possible rational roots are \(\pm 1, \pm 2, \pm \frac{1}{3}, \pm \frac{2}{3}\).

Step 2: Test Possible Rational Roots

Substitute each possible rational root into the equation to see if it equals zero. Start with \(x = 1\):1. \(3(1)^3 + 2(1)^2 - 3(1) - 2 = 3 + 2 - 3 - 2 = 0\)2. Thus, \(x = 1\) is a root.Next, factor \(x - 1\) out from the cubic equation.

Step 3: Factor the Cubic Polynomial

Using the fact that \(x = 1\) is a root, perform synthetic division or polynomial division on \(3x^3 + 2x^2 - 3x - 2\) by \(x - 1\). This reduces the polynomial:After the division, the quotient is \(3x^2 + 5x + 2\). So, \(3x^3 + 2x^2 - 3x - 2 = (x - 1)(3x^2 + 5x + 2)\).

Step 4: Solve the Quadratic Equation

Solve the quadratic equation \(3x^2 + 5x + 2 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 3\), \(b = 5\), and \(c = 2\):1. Calculate the discriminant: \(b^2 - 4ac = 5^2 - 4 \times 3 \times 2 = 25 - 24 = 1\).2. Substitute into the quadratic formula: \[x = \frac{-5 \pm \sqrt{1}}{6} = \frac{-5 \pm 1}{6}.\]3. Thus, the solutions are \(x = -\frac{4}{6} = -\frac{2}{3}\) and \(x = -\frac{6}{6} = -1\).

Step 5: Write All Solutions

Combine the solutions found from both steps. The solutions to the equation \(3x^3 + 2x^2 - 3x - 2 = 0\) are \(x = 1\), \(x = -\frac{2}{3}\), and \(x = -1\). None of these solutions are complex numbers, so there is no need to consider complex conjugates for this equation.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a useful tool for finding potential rational solutions of polynomial equations. It tells us that if a polynomial has a rational root, that root must be a factor of the constant term divided by a factor of the leading term. In our polynomial, \(3x^3 + 2x^2 - 3x - 2\), the constant term is \(-2\) and the leading coefficient is \(3\).

Thus, the possible rational roots are:

• \(\pm 1\)
• \(\pm 2\)
• \(\pm \frac{1}{3}\)
• \(\pm \frac{2}{3}\)

By testing these potential roots, we narrow down the possibilities to find the actual rational root. In our equation, \(x = 1\) is verified as a root by substitution, simplifying the equation further.

Quadratic Equation

A quadratic equation is any polynomial equation of the form \(ax^2 + bx + c = 0\). The quadratic equation in our exercise was obtained after factoring the original cubic polynomial via division: \(3x^2 + 5x + 2 = 0\).

To solve this quadratic equation, we utilize the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, \(a = 3\), \(b = 5\), and \(c = 2\). Solving this provides us with two distinct solutions: \(x = -\frac{2}{3}\) and \(x = -1\). The use of the quadratic formula is essential in finding the roots of quadratics, especially when factoring is not straightforward.

Synthetic Division

Synthetic division is a shorthand method of polynomial division, particularly useful when dividing by linear factors like \(x - 1\). Compared to long division, synthetic division streamlines the process, employing only the coefficients of the polynomial.

To perform synthetic division for our polynomial \(3x^3 + 2x^2 - 3x - 2\) by \(x - 1\):

• Set up the coefficients: \([3, 2, -3, -2]\)
• Use the root \(x = 1\) as the divisor
• Bring down the first coefficient \(3\).
• Continue through the process, multiplying and adding down.

After completing synthetic division, the quotient is \(3x^2 + 5x + 2\). This result gives us a new quadratic polynomial to solve. This method is efficient and reduces the effort in dividing polynomials.

Polynomial Division

Polynomial division is a technique by which polynomials are divided, one term at a time, similar to long division with numbers. It's essential for simplifying polynomials and finding factors.

In this context, polynomial division helps us reduce our given polynomial \(3x^3 + 2x^2 - 3x - 2\) by \(x - 1\), a known root, to obtain \(3x^2 + 5x + 2\).

Here's a quick refresher on the steps involved in polynomial division:

• Divide the first term of the dividend by the first term of the divisor.
• Multiply the entire divisor by this result.
• Subtract that from the original polynomial.
• Repeat with the new resulting polynomial.

By methodically applying these steps, polynomial division simplifies complex polynomials into more manageable components, a critical step in solving higher order equations.