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Chapter 2: Problem 5

The sum of the roots of \(3 x^{3}+4 x^{2}-4 x=0\) is (A) \(-\frac{4}{3}\) (B) \(^{-\frac{3}{4}}\) (C) 0 (D) \(\frac{4}{3}\) (E) 4

Short Answer

Expert verified
The sum of the roots of the equation \(3x^3 + 4x^2 - 4x = 0\) is \(-4/3\) so it corresponds to option (A).

Step by step solution

01

1. Simplify the Equation

Firstly, it can be observed that each term of the equation \(3x^3 + 4x^2 - 4x = 0\) has a common factor, \(x\). Therefore, this equation can be simplified by factoring out \(x\), which then gives us the equation \(x(3x^2 + 4x - 4) = 0\).
02

2. Identify the Roots

The cubic equation \(x(3x^2 + 4x - 4) = 0\) has its roots when \(x = 0\) or when \(3x^2 + 4x - 4 = 0\). While \(x = 0\) is an obvious root, the other roots can be obtained after applying the quadratic formula on the equation \(3x^2 + 4x - 4 =0\).
03

3. Apply the Quadratic Formula

Applying the quadratic formula, \(x = [-b 卤 sqrt(b^2 - 4ac)]/2a\), on the quadratic part of the equation \(3x^2 + 4x - 4 = 0\), where a = 3, b = 4, and c = -4, we obtain two roots. Those roots are \((-4 卤 sqrt((4)^2 - 4*3*(-4)))/2*3 = (-4 卤 sqrt(16 + 48))/6 = (-4 卤 sqrt(64))/6 = (-4 卤 8)/6\) which simplifies down to \(x = 2/3\) and \(x = -2\).
04

4. Calculate the Sum of Roots

The sum of all roots of the equation \(x(3x^2 + 4x - 4) = 0\) is obtained by adding the roots \(x = 0\), \(x = 2/3\), and \(x = -2\), which equals \(0 + 2/3 - 2 = -4/3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cubic Equation
A cubic equation is a polynomial equation of degree three. In simpler words, it includes terms until the power of three, making it the highest degree term. These equations are generally written in the form \[ ax^3 + bx^2 + cx + d = 0 \]where \(a\), \(b\), \(c\), and \(d\) are constants, and \(x\) represents the variable. They're fundamental in algebra, providing insights into the characteristics and behaviors of many functions.
  • Example: An example of a cubic equation would be \(3x^3 + 4x^2 - 4x = 0\).
  • The highest power in this equation is 3, making it a cubic equation.
Cubic equations can have up to three real roots, or solutions, because the value of \(x\) can make the equation equal to zero in three unique situations.
Quadratic Formula
The quadratic formula is a method used for finding the roots of a quadratic equation. A quadratic equation is in the form \[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are coefficients, and \(x\) is the unknown variable.
  • The formula can be expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
  • It is particularly useful when other factoring methods are cumbersome or not apparent.
In the context of our cubic equation, when we factor out a common term, we arrive at a quadratic equation which can be solved using the quadratic formula. This provides us with two additional roots beyond the obvious one from the original cubic equation.
Factoring Polynomials
Factoring is a mathematical process used to simplify polynomials by expressing them as a product of simpler polynomials. With cubic equations, this often means pulling out a common variable or term to break the equation into an easier form.
  • Example: In the equation \(3x^3 + 4x^2 - 4x = 0\), we can factor out an \(x\) from each term.
  • This results in \(x(3x^2 + 4x - 4) = 0\), reducing the cubic equation into a simpler polynomial factor times a quadratic equation.
Factoring is essential because it turns complex polynomial equations into manageable parts, thereby making it easier to solve for roots. It simplifies the equation to a point where other solving techniques, like the quadratic formula, can be applied.
Polynomial Roots
Polynomial roots are the solutions to the equation \(P(x) = 0\). They are the values of \(x\) that make the polynomial equal to zero. Understanding how to find these roots is key to solving polynomial equations.
  • For the cubic equation \(3x^3 + 4x^2 - 4x = 0\), we first factored to find that one root is \(x = 0\).
  • The remaining quadratic aspect \(3x^2 + 4x - 4 = 0\) gave us two additional roots when using the quadratic formula: \(x = \frac{2}{3}\) and \(x = -2\).
The roots of a polynomial give insights into its behavior on a graph, indicating where the function crosses the x-axis. By finding the roots, we can effectively solve these equations, understand their characteristics, such as the sum of roots, which in this case is \(-\frac{4}{3}\). Polynomials, especially cubic ones, can have multiple roots, making the understanding of their solutions both intricate and illuminating.

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