Problem 19 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$x^{4}+3 x^{3}-30 x^{2}-6 x+56=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 2, x = -2, x = \frac{-3 + \sqrt{65}}{2}, x = \frac{-3 - \sqrt{65}}{2}\).

Step 1: Identify Possible Rational Roots

To find potential roots of the polynomial, use the Rational Root Theorem. The factors of the constant term \(56\) are \(\pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56\). The leading coefficient is \(1\), so the possible rational roots are these factors.

Step 2: Test Rational Roots

Substitute the potential rational roots \(\pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56\) into the polynomial to determine which are actual roots. After testing, both \(x = 2\) and \(x = -2\) are found to be roots of the polynomial.

Step 3: Polynomial Division

Perform synthetic division on the original polynomial by \((x - 2)\) and \((x + 2)\) to reduce the degree of the polynomial. After dividing by both roots, the polynomial reduces to \(x^2 + 3x - 14 = 0\).

Step 4: Solve the Quadratic Equation

Now solve the quadratic equation \(x^2 + 3x - 14 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 3\), and \(c = -14\). Calculate the discriminant: \(b^2 - 4ac = 3^2 - 4 \times 1 \times (-14) = 9 + 56 = 65\). Thus, \(x = \frac{-3 \pm \sqrt{65}}{2}\).

Step 5: Collect All Solutions

From the previous steps, the solutions to the original polynomial equation are \(x = 2, x = -2, x = \frac{-3 + \sqrt{65}}{2}, x = \frac{-3 - \sqrt{65}}{2}\). These are the solutions for the equation \(x^4 + 3x^3 - 30x^2 - 6x + 56 = 0\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a fantastic tool for solving polynomial equations, especially when working with higher-degree polynomials like quartics or cubics. This theorem helps us to identify all possible rational roots of a polynomial equation.
It states that any potential rational root of a polynomial equation has the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient.
For the given polynomial, \( x^4 + 3x^3 - 30x^2 - 6x + 56 = 0 \), the constant term is 56 and the leading coefficient is 1. Therefore, the possible rational roots simplify to the factors of 56.
These factors are:

• \( \pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56 \)

By substituting these values into the polynomial, one can identify the actual roots without having to graph the function.

Polynomial Division

Polynomial division is a method used to simplify polynomial equations once one or more roots are discovered. In this exercise, after finding that \( x = 2 \) and \( x = -2 \) are roots, polynomial division helps reduce the polynomial to a simpler form.
Synthetic division is often the method of choice due to its efficiency and ease. It involves less writing, especially when the leading coefficient is one, as it is here.
The polynomial \( x^4 + 3x^3 - 30x^2 - 6x + 56 = 0 \) initially divides by \( (x-2) \) and \( (x+2) \). After each division, the polynomial’s degree decreases.
Ultimately, it simplifies to \( x^2 + 3x - 14 = 0 \), making the polynomial equation much easier to solve.

Quadratic Formula

Once a polynomial is reduced to a quadratic form, as in the case of the equation \( x^2 + 3x - 14 = 0 \), the quadratic formula is the solution method of choice.
The formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

allows us to find the roots of any quadratic equation, where \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term.
For our equation, we have \( a = 1 \), \( b = 3 \), and \( c = -14 \). Substituting these values into the quadratic formula gives us the roots as follows:

• \( x = \frac{-3 \pm \sqrt{65}}{2} \)

These roots can be written in a more exact form or approximated as needed.

Discriminant

The discriminant is a core part of the quadratic formula and provides insights into the nature of a quadratic equation's roots. It is the part under the square root sign: \( b^2 - 4ac \).
For our quadratic \( x^2 + 3x - 14 = 0 \), the discriminant calculates as follows:

• \( b^2 - 4ac = 3^2 - 4 \times 1 \times (-14) = 9 + 56 = 65 \)

A positive discriminant, like 65, indicates that the quadratic equation has two distinct real roots.
This information can help predict the nature of the solutions and whether additional transformations or methods are necessary to find all roots of the polynomial equation.