Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(39-56:\) Find the solutions of the equation. $$ 8 x^{3}-12 x^{2}+2 x-3=0 $$

Step-by-Step Solution

Answer

The roots are \(x = 1\), \(x = \frac{1 + \sqrt{7}}{4}\), \(x = \frac{1 - \sqrt{7}}{4}\).

Step 1: Analyze the Equation

The equation given is a cubic polynomial: \(8x^3 - 12x^2 + 2x - 3 = 0\). Our goal is to find the roots of this cubic equation, which are values of \(x\) that satisfy the equation.

Step 2: Initial Checks for Possible Rational Roots

We use the Rational Root Theorem to identify possible rational roots. The potential rational roots can be \(\pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{1}{4}, \pm \frac{3}{4}, \pm 8, \pm \frac{3}{8} \). We will check these values to determine if any are roots of the polynomial.

Step 3: Test Potential Roots using Synthetic Division

To verify, we can test \(x = 1\):- Coefficients of the polynomial are \(8, -12, 2, -3\).- Perform synthetic division on \(8x^3 - 12x^2 + 2x - 3\) by \(x - 1\).- The result is: (8 \, -4 \, -2 \, 1), with a remainder of 0.This confirms \(x = 1\) is a root.

Step 4: Factor the Cubic Polynomial

Since \(x=1\) is a root, \(x-1\) is a factor. Using synthetic division, we factor \(8x^3 - 12x^2 + 2x - 3\) as \((x - 1)(8x^2 - 4x - 3)\).

Step 5: Solve the Quadratic Factor

Now, solve \(8x^2 - 4x - 3 = 0\) using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Where \(a = 8\), \(b = -4\), and \(c = -3\).Compute the discriminant: \((-4)^2 - 4\times8\times(-3) = 16 + 96 = 112\).Find the roots: \[x = \frac{4 \pm \sqrt{112}}{16}\]\[x = \frac{4 \pm 4\sqrt{7}}{16}\]\[x = \frac{1 \pm \sqrt{7}}{4}\]

Step 6: Conclude with the Set of Solutions

The solutions to the original equation \(8x^3 - 12x^2 + 2x - 3 = 0\) are:- \(x = 1\)- \(x = \frac{1 + \sqrt{7}}{4}\)- \(x = \frac{1 - \sqrt{7}}{4}\).

Key Concepts

Rational Root Theorem

When dealing with polynomial equations like cubic ones, the Rational Root Theorem is a valuable tool. It helps us find potential rational solutions or roots for these equations. In simple terms, the theorem states that any rational root, i.e., a solution in the form of a fraction \( \frac{p}{q} \), will have a numerator \( p \) that is a factor of the constant term, and a denominator \( q \) that is a factor of the leading coefficient.
For the equation \( 8x^3 - 12x^2 + 2x - 3 = 0 \), our constant term is \(-3\), and the leading coefficient (the coefficient of \( x^3 \)) is \(8\).

• The factors of \(-3\) are \( \pm 1, \pm 3 \).
• The factors of \(8\) are \( \pm 1, \pm 2, \pm 4, \pm 8 \).

By pairing these factors, we get potential rational roots: \( \pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{1}{4}, \pm \frac{3}{4}, \pm \frac{1}{8}, \pm \frac{3}{8} \). These are the candidates we'll test to see if they make the polynomial equal zero.

Synthetic Division

Synthetic division is a shortcut method to divide a polynomial by a binomial of the form \( x - c \), and it simplifies the process compared to long division. This technique is particularly useful in verifying potential roots found using the Rational Root Theorem.
To perform synthetic division:

• Write down the coefficients of the polynomial. For \( 8x^3 - 12x^2 + 2x - 3 \), they are \( 8, -12, 2, -3 \).
• Choose a candidate from the Rational Root Theorem. Let's test \( x = 1 \).
• Drop the leading coefficient (8) to the bottom row and then multiply it by your candidate (1). Add the result to the next coefficient (-12) and continue the process.

The results of synthetic division when using \( x = 1 \) leads to a remainder of zero, confirming that 1 is indeed a root. The synthetic division also provides the coefficients of the quotient polynomial, which is \( 8x^2 - 4x - 3 \).

Quadratic Formula

After using synthetic division to verify a rational root, we factored the cubic polynomial into \( (x - 1)(8x^2 - 4x - 3) \). Our task now is to solve the quadratic part \( 8x^2 - 4x - 3 = 0 \) using the quadratic formula.
The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 8 \), \( b = -4 \), and \( c = -3 \). The formula helps find the roots of any quadratic equation by calculating their values directly.
First, compute the discriminant \( b^2 - 4ac \):
It becomes \( (-4)^2 - 4 \times 8 \times (-3) = 16 + 96 = 112 \). This positive discriminant indicates that there are two real roots.
Substitute back into the formula:

• \( x = \frac{4 \pm \sqrt{112}}{16} \)
• Simplifying, we get \( x = \frac{1 \pm \sqrt{7}}{4} \).

These solutions, along with \( x = 1 \), are the roots of the original cubic equation.