Problem 98 - Algebra and Trigonometry

Question

Solve the equation \(2 x^{3}+5 x^{2}-4 x-3=0\) given that \(-3\) is a zero of \(f(x)=2 x^{3}+5 x^{2}-4 x-3\)

Step-by-Step Solution

Answer

The roots of the equation \(2 x^{3}+5 x^{2}-4 x-3=0\) are \(x = -3\), \(x = 1\), and \(x = -0.5\).

Step 1: Apply Synthetic Division

With -3 as a root, synthetic division is performed on the cubic function, resulting in a process that gives: \[\begin{{array}}{{cccc}}-3 & 2 & 5 & -4 & -3 \ & & -6 & 3 & 3 \\hline & 2 & -1 & -1 & 0\end{{array}}\]The coefficients of the resulting polynomial are 2, -1, and -1.

Step 2: Form And Solve Quadratic Equation

From the synthetic division, the quadratic equation is formed as: \(2x^{2} - x - 1 = 0\). To solve this quadratic equation apply the quadratic formula:\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)Where \(a = 2\), \(b = -1\), and \(c = -1\). Substituting these values yields:\(x = \frac{1 \pm \sqrt{1+8}}{4}\)After solving the expression, we find that \(x = 1, -0.5\) are the roots of the equation.

Step 3: Validate the Solutions

Insert the values \(x = -3, 1, -0.5\) into the original cubic function. If the results equal to 0, that confirms the given values are indeed roots to the equation

Key Concepts

Understanding Synthetic Division

Synthetic division is a handy method for dividing polynomials when we know one of the roots or zeros of the polynomial function. This method simplifies the process compared to traditional long division.
You start by writing the coefficients of the polynomial in descending order of power. In the example, the polynomial is \(2x^3 + 5x^2 - 4x -3\), so we list the coefficients: \(2, 5, -4, -3\).
Next, we use the given root, \(-3\), and place it outside the synthetic division setup. Here's a quick step-by-step guide:

• Bring down the first coefficient (2 in this case) to start the process.
• Multiply this coefficient by the root, \(-3\), and add the result to the next coefficient.
• Continue this process for each coefficient. The value at the last step should be zero if \(-3\) is indeed a root.

The resulting new coefficients, \(2, -1, -1\), form a new polynomial \(2x^2 - x - 1\), which can be further solved. Synthetic division ensures efficiency, especially for higher-degree polynomials.

Demystifying the Quadratic Formula

The quadratic formula is a universal tool used to find the roots of a quadratic equation. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). The roots can be found using: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's how it works:

• Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation. In the problem, these are \(a = 2\), \(b = -1\), and \(c = -1\).
• Substitute these values into the formula.
• Calculate the discriminant \((b^2 - 4ac)\) to determine the nature of the roots.
• Solve the equation. The plus-minus sign (\(\pm\)) indicates there can be two roots.

In practical terms, the quadratic formula here gives the roots \(x = 1\) and \(x = -0.5\). This formula is unbeatable for delivering accurate solutions efficiently.

Exploring Cubic Functions

Cubic functions are a category of polynomial equations with the highest degree of three. A general cubic function has the form \(ax^3 + bx^2 + cx + d = 0\). These equations can have up to three real roots, and possibly complex ones.
In a cubic function, solving might start with finding one real root, often discovered through testing small integer values or given in problems, like in the example where \(\(-3\)\) is a known root.
Once a root is known, polynomial division, often through synthetic division, is used to reduce the cubic equation to a quadratic form. From the remainder, we can more easily solve for any additional roots. Here, once reduced to \(2x^2 - x - 1\), we applied the quadratic formula to find the other roots.
Cubic functions are fundamental in mathematics, as they model various real-world phenomena from physics to engineering, often representing rates of change or various spatial dimensions.