Problem 20 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$3 x^{3}-10 x^{4}-6 x^{3}+24 x^{2}+11 x-6=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 2, -3\), and the roots of the remaining quadratic equation.

Step 1: Simplify the Equation

Combine like terms in the equation. Start by noticing that the terms \(3x^3\) and \(-6x^3\) can be combined. This simplifies the equation to \(-10x^4 -3x^3 +24x^2 +11x -6=0\).

Step 2: Identify and Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of the polynomial equation \(a_nx^n + a_{n-1}x^{n-1} + \, ... \, + a_0 = 0\) is of the form \(\pm \frac{p}{q}\), where \(p\) is a factor of the constant term \(a_0\) and \(q\) is a factor of the leading coefficient \(a_n\). For this equation, the possible rational roots are \(\pm 1, \pm 2, \pm 3, \pm 6\).

Step 3: Test Potential Roots

Substitute each potential rational root into the polynomial to see if any result in zero:- For \(x = 1\): \(-10(1)^4 - 3(1)^3 + 24(1)^2 + 11(1) - 6 = -10 - 3 + 24 + 11 - 6 = 16 eq 0\).- Continue testing other candidates similarly. \(x = 2\) is a root: \(-10(2)^4 - 3(2)^3 + 24(2)^2 + 11(2) - 6 = 0\).

Step 4: Perform Polynomial Division

Since \(x = 2\) is a root, we can divide the polynomial \(-10x^4 -3x^3 +24x^2 +11x -6\) by \(x-2\) to find the other factors. Use synthetic or long division to obtain a quotient polynomial. The quotient is \(-10x^3 - 23x^2 - 22x - 3\).

Step 5: Factor the Quotient

Attempt to factor \(-10x^3 - 23x^2 - 22x - 3\). Continue using methods such as synthetic division or trial roots from the Rational Root Theorem to find additional roots and factor completely. Upon further analysis, \(x = -3\) is another root.

Step 6: Solve the Remaining Quadratic

After factoring out \(x + 3\) from \(-10x^3 - 23x^2 - 22x - 3\), we have a quadratic equation. Solve this quadratic using the quadratic formula or factoring. The quadratic coefficients yield the final solution set.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool in solving polynomial equations. It helps identify potential rational roots, making the process more efficient. The theorem states that for a polynomial equation \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0\), any potential rational root \(\frac{p}{q}\) must have \(p\) as a factor of the constant term \(a_0\) and \(q\) as a factor of the leading coefficient \(a_n\).

• Identify the constant term, \(a_0\).
• Determine the leading coefficient, \(a_n\).
• List all factors of \(a_0\) and \(a_n\).
• Form possible rational roots by taking \(\pm \frac{p}{q}\).

In the original exercise, the Rational Root Theorem narrowed down potential solutions to \(\pm 1, \pm 2, \pm 3, \pm 6\). This smart-cut reduces the number of trials needed to test compared to random guesses. By substituting these into the polynomial, we efficiently check if they satisfy the equation, simplifying our solving process.

Polynomial Division

Polynomial division is much like long division with numbers, but instead, it involves variables. After identifying a root using the Rational Root Theorem, polynomial division helps simplify the polynomial by reducing its degree. This is important because lower degree polynomials are easier to manage and solve.When you know a root, say \(x = 2\), divide the polynomial by \(x - 2\). You can use either synthetic division or long division. Synthetic division is generally quicker for linear divisors.

• Set up for synthetic division using the root you found.
• Carry out the division step-by-step to get a new polynomial.
• The resulting polynomial reflects the factors and remaining equations to solve.

For the exercise, dividing the original polynomial by \(x - 2\) resulted in the simpler cubic polynomial \(-10x^3 - 23x^2 - 22x - 3\). This process distills the complex polynomial into more manageable parts for further factorization or root evaluation.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler components, or factors, that multiply to give the original polynomial. This can be essential in finding all of its roots. After conducting polynomial division, you simplify the polynomial, identifying additional factorable portions.

• Use the Rational Root Theorem to continue testing for factors.
• Identify any common patterns, such as difference of squares or grouping.
• For trinomials or higher-degree polynomials, look for roots through testing or use known patterns.

In the exercise, after dividing by \(x = 2\) and obtaining a cubic equation, the additional root \(x = -3\) was recognized, which helped in further factorization. This step-by-step breakdown aids in identifying the small building blocks needed to solve complex polynomials.

Quadratic Formula

The Quadratic Formula is a reliable method to find roots of quadratic equations, especially when they cannot be easily factored. For a quadratic equation in the form \(ax^2 + bx + c = 0\), the formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here’s how you apply it:

• Identify coefficients \(a\), \(b\), and \(c\) from the equation.
• Calculate the discriminant \(b^2 - 4ac\).
• Use the quadratic formula to find \(x\), with two solutions possible due to the \(\pm\) symbol.

In solving the polynomial in the exercise, once a remaining quadratic was isolated from the earlier steps, the Quadratic Formula enabled finding the precise roots. This method is particularly useful when roots are irrational or complex, ensuring that no possible solutions are missed.