Find all solutions of the following polynomial equation.

31x−18x^2−9x^3+12=0

We are given the polynomial equation:

\[
31x - 18x^2 - 9x^3 + 12 = 0
\]

To find the solutions, we'll follow these steps:

### Step 1: Rearrange the equation
Rearrange the terms in descending order of powers of \( x \):

\[
-9x^3 - 18x^2 + 31x + 12 = 0
\]

### Step 2: Try Rational Root Theorem
The Rational Root Theorem helps us find possible rational roots by considering factors of the constant term (12) and the leading coefficient (-9).

The factors of \( 12 \) are: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)

The factors of \( -9 \) are: \( \pm 1, \pm 3, \pm 9 \)

The possible rational roots are:

\[
\pm 1, \pm \frac{1}{3}, \pm \frac{1}{9}, \pm 2, \pm \frac{2}{3}, \pm 3, \pm 4, \pm \frac{4}{3}, \pm 6, \pm 12
\]

### Step 3: Test the possible roots
We will test these possible roots by substituting them into the polynomial.

Let's start with \( x = 1 \):

\[
-9(1)^3 - 18(1)^2 + 31(1) + 12 = -9 - 18 + 31 + 12 = 16 \neq 0
\]

Now, \( x = -1 \):

\[
-9(-1)^3 - 18(-1)^2 + 31(-1) + 12 = 9 - 18 - 31 + 12 = -28 \neq 0
\]

Next, let's try \( x = 2 \):

\[
-9(2)^3 - 18(2)^2 + 31(2) + 12 = -9(8) - 18(4) + 62 + 12 = -72 - 72 + 62 + 12 = -70 \neq 0
\]

I'll test more rational roots.From testing the possible rational roots, we found two roots that make the polynomial zero:

- \( x = -3 \)
- \( x = \frac{4}{3} \)

### Step 4: Polynomial division
Now, we divide the original polynomial \( -9x^3 - 18x^2 + 31x + 12 \) by \( (x + 3) \) to find the quotient and further factor the polynomial.

Let's proceed with polynomial division to find the remaining factor.After dividing the polynomial by \( (x + 3) \), the quotient is:

\[
-9x^2 + 9x + 4
\]

Now, we solve the quadratic equation:

\[
-9x^2 + 9x + 4 = 0
\]

### Step 5: Solve the quadratic equation
We can apply the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For \( -9x^2 + 9x + 4 = 0 \), the coefficients are \( a = -9 \), \( b = 9 \), and \( c = 4 \). Let's solve it.The solutions to the quadratic equation \( -9x^2 + 9x + 4 = 0 \) are:

\[
x = -\frac{1}{3} \quad \text{and} \quad x = \frac{4}{3}
\]

### Step 6: Summary of all solutions
The complete set of solutions to the polynomial equation \( 31x - 18x^2 - 9x^3 + 12 = 0 \) is:

\[
x = -3, \quad x = \frac{4}{3}, \quad x = -\frac{1}{3}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How does the Rational Root Theorem help in solving polynomials?
2. Can polynomial division always be used to simplify higher-degree equations?
3. How do you apply the quadratic formula in general?
4. Why are there no more than three solutions for a cubic equation?
5. What are the steps to factor polynomials more efficiently?

**Tip:** Always verify your solutions by substituting them back into the original equation.

Find all solutions of the following polynomial equation: 31x - 18x^2 - 9x^3 + 12 = 0

Learn how to solve the cubic polynomial equation 31x - 18x^2 - 9x^3 + 12 = 0 step-by-step using the Rational Root Theorem, polynomial division, and the quadratic formula. This guide is suitable for high school students (Grades 10-12) and covers important algebraic methods for solving higher-degree equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation