Problem 17 - Precalculus: Functions and Graphs

Question

Find all solutions of the equation. $$2 x^{3}-3 x^{2}-17 x+30=0$$

Step-by-Step Solution

Answer

The solutions are \( x = 3 \), \( x = \frac{-3 + \sqrt{89}}{4} \), and \( x = \frac{-3 - \sqrt{89}}{4} \).

Step 1: Identify the Type of Equation

The given equation is a cubic polynomial equation of the form \( ax^3 + bx^2 + cx + d = 0 \). In this case, \( a = 2 \), \( b = -3 \), \( c = -17 \), and \( d = 30 \).

Step 2: Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational solution, \( \frac{p}{q} \), must be such that \( p \) is a factor of the constant term (30), and \( q \) is a factor of the leading coefficient (2). Thus, the possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{5}{2}, \pm \frac{15}{2} \).

Step 3: Test Possible Rational Roots

By substituting each possible rational root into the polynomial, we find that \( x = 3 \) satisfies \( 2x^3 - 3x^2 - 17x + 30 = 0 \), because it results in a value of 0. Thus, \( x = 3 \) is a root.

Step 4: Perform Synthetic Division

To find the remaining roots, perform synthetic division of \( 2x^3 - 3x^2 - 17x + 30 \) by \( x - 3 \). The division gives the quotient \( 2x^2 + 3x - 10 \), which means \( 2x^3 - 3x^2 - 17x + 30 = (x - 3)(2x^2 + 3x - 10) \).

Step 5: Solve the Quadratic Equation

Solve the quadratic equation \( 2x^2 + 3x - 10 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = 2 \), \( b = 3 \), and \( c = -10 \) to get \( x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-10)}}{4} = \frac{-3 \pm \sqrt{89}}{4} \).

Step 6: State the Solutions

The solutions to the equation \( 2x^3 - 3x^2 - 17x + 30 = 0 \) are \( x = 3 \), \( x = \frac{-3 + \sqrt{89}}{4} \), and \( x = \frac{-3 - \sqrt{89}}{4} \). These are all the roots of the equation.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a helpful tool when tackling polynomial equations. It tells us about any potential rational solutions of the equation. It's especially useful for polynomials with integer coefficients. According to the theorem, the potential rational roots of a polynomial are in the form \( \frac{p}{q} \). Here, \( p \) is a factor of the constant term at the end of the polynomial, and \( q \) is a factor of the leading coefficient.
In our example, the polynomial \( 2x^3 - 3x^2 - 17x + 30 \) has a constant term of 30 and a leading coefficient of 2. Therefore, all factors of 30, such as ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30, combined with factors of 2, ±1 and ±2, give us a list of possible rational roots.
The idea is to test these potential roots in the polynomial to see which ones actually satisfy the equation (make it equal to zero). In this case, through substitution, we find that \( x = 3 \) is indeed a root.

Synthetic Division

Synthetic division is a straightforward and efficient method for dividing a polynomial by a binomial of the form \( x - c \). It simplifies the long division process and is especially useful for finding polynomial roots.
After identifying \( x = 3 \) as a root using the Rational Root Theorem, we applied synthetic division to \( 2x^3 - 3x^2 - 17x + 30 \) with \( x - 3 \). This means we divided the polynomial by \( x - 3 \), expecting a quotient that is a simpler polynomial.
The result of this division was \( 2x^2 + 3x - 10 \), indicating that the original polynomial can be factored as \( (x - 3)(2x^2 + 3x - 10) \). This process helps break down the polynomial into more manageable parts, allowing for further root solving.

Quadratic Formula

To find the roots of the quadratic polynomial \( 2x^2 + 3x - 10 \) that resulted from the synthetic division, we use the quadratic formula. This formula is a foolproof way to find solutions to any quadratic equation, \( ax^2 + bx + c = 0 \).
The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 2 \), \( b = 3 \), and \( c = -10 \). Plugging these values into the formula gives us the solutions \( x = \frac{-3 + \sqrt{89}}{4} \) and \( x = \frac{-3 - \sqrt{89}}{4} \).
This method relies on the discriminant, \( b^2 - 4ac \), to determine the nature of the roots. In our case, the values under the square root are positive, implying two distinct real roots.

Polynomial Roots

Finding the roots of a polynomial is crucial because these roots are the solutions to the polynomial equation, where the equation evaluates to zero. In our example, the polynomial \( 2x^3 - 3x^2 - 17x + 30 = 0 \) is a cubic polynomial, which can have up to three real roots.
Through the combination of the Rational Root Theorem, synthetic division, and the quadratic formula, we identified all possible roots.
The solutions found are \( x = 3 \), \( x = \frac{-3 + \sqrt{89}}{4} \), and \( x = \frac{-3 - \sqrt{89}}{4} \). Each root represents a point where the graph of the polynomial would intersect the x-axis. Understanding how to find these roots is essential for analyzing polynomials and their graphs.