Problem 18 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$12 x^{3}+8 x^{2}-3 x-2=0$$

Step-by-Step Solution

Answer

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

Step 1: Check for Rational Root

To find rational roots, use the Rational Root Theorem, which suggests possible roots are factors of the constant term (-2) divided by factors of the leading coefficient (12). This gives possible roots: \( \pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{12} \). Check these values in the polynomial.

Step 2: Evaluate Possible Roots

Test potential roots by substituting each value into \(12x^3 + 8x^2 - 3x - 2 = 0\). Substitute \(x = -\frac{1}{2}\) and find it satisfies the equation. Thus, \(x = -\frac{1}{2}\) is a root.

Step 3: Factor the Polynomial

Using polynomial division, divide \(12x^3 + 8x^2 - 3x - 2\) by \(x + \frac{1}{2}\). The remainder is zero, confirming \(x + \frac{1}{2}\) as a factor. The quotient is \(12x^2 + 2x - 4\).

Step 4: Solve the Quadratic Equation

Solve \(12x^2 + 2x - 4 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 12\), \(b = 2\), and \(c = -4\). Calculate the discriminant \(b^2 - 4ac = 4 + 192 = 196\). The solutions are \(x = \frac{-2 \pm \sqrt{196}}{24}\).

Step 5: Calculate the Solutions

Since \(\sqrt{196} = 14\), the roots are \(x = \frac{-2 + 14}{24} = \frac{1}{2}\) and \(x = \frac{-2 - 14}{24} = -\frac{2}{3}\). Therefore, the solutions to the quadratic are \(x = \frac{1}{2}\) and \(x = -\frac{2}{3}\).

Step 6: Compile All Solutions

The complete set of solutions for the polynomial equation \(12x^3 + 8x^2 - 3x - 2 = 0\) are \(x = -\frac{1}{2}, \frac{1}{2}, \text{and} -\frac{2}{3}\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a helpful tool when dealing with polynomial equations. It provides a list of possible rational roots based on the factors of the constant term and the leading coefficient.
In our example, the polynomial is \(12x^3 + 8x^2 - 3x - 2=0\). The constant term here is -2 while the leading coefficient is 12.
By considering the factors of -2 (+-1, +-2) and factors of 12 (+-1, +-2, +-3, +-4, +-6, +-12), we can write out all possible rational roots as fractions:

• \( \pm 1 \)
• \( \pm 2 \)
• \( \pm \frac{1}{2} \)
• \( \pm \frac{1}{3} \)
• \( \pm \frac{1}{4} \)
• \( \pm \frac{1}{6} \)
• \( \pm \frac{1}{12} \)

This theorem doesn’t tell us which numbers definitely are roots, only which might be. Checking each possibility by substituting it back into the polynomial allows us to find actual roots such as \(x = -\frac{1}{2}\).

Polynomial Division

Once a rational root is found, polynomial division helps to break down the polynomial further.
In our problem, after discovering that \(x = -\frac{1}{2}\) is a root of the polynomial \(12x^3 + 8x^2 - 3x - 2\), we use polynomial division to divide the polynomial by the factor \(x + \frac{1}{2}\).
This is similar to long division you learned in basic math, but applied to algebraic terms.

• Divide: Determine how many times the first term of the divisor (\(x\)) goes into the first term of the dividend (\(12x^3\)).
• Multiply: Multiply the entire divisor by this term, and place the product under the dividend.
• Subtract: Subtract these terms from the original dividend.
• Repeat: Carry down the next term and repeat until completely divided.

The result of this division is a new polynomial, \(12x^2 + 2x - 4\), which leads us to solving a quadratic equation.

Quadratic Formula

The Quadratic Formula is essential for solving quadratic equations like the one we derive from our polynomial division, \(12x^2 + 2x - 4 = 0\).
It provides a straightforward way to find roots of any quadratic equation. The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This applies to equations of the form \(ax^2 + bx + c = 0\). Here, \(a = 12\), \(b = 2\), and \(c = -4\).
The formula relies on calculating the discriminant \(b^2 - 4ac\), ensuring the solutions are within the reach of possible real roots. For our example, plug the values into the quadratic formula and solve for \(x\).

Discriminant Calculation

The discriminant is a key part of understanding the nature of roots found with the Quadratic Formula.
The formula for the discriminant is \(b^2 - 4ac\). It determines whether the roots of a quadratic equation are real or complex.
Consider our example from the quadratic part:

• Compute \(b^2\) which results in \(2^2 = 4\).
• Calculate \(4ac\) as \(4 \times 12 \times (-4) = -192\).
• The entire discriminant calculation becomes: \(4 - (-192) = 196\).

A positive discriminant, like 196, indicates two distinct real roots.
These roots are calculated through \(x = \frac{-2 \pm \sqrt{196}}{24}\), resulting in \(x = \frac{1}{2}\) and \(x = -\frac{2}{3}\).
Understanding the discriminant helps predict the type and number of solutions without solving the full equation directly.