Problem 40 - College Algebra

Question

Solve the equation \(2 x^{3}-3 x^{2}-11 x+6=0\) given that \(-2\) is a zero of \(f(x)=2 x^{3}-3 x^{2}-11 x+6\).

Step-by-Step Solution

Answer

The solutions to the equation \(2 x^{3}-3 x^{2}-11 x+6=0\) are \(-2\) and the solutions of the quadratic equation derived from the polynomial division.

Step 1: Polynomial Division

Firstly, divide the polynomial \(2 x^{3} - 3 x^{2} - 11 x + 6\) by \(x + 2\). This can be done using polynomial division. The result will be a quadratic expression.

Step 2: Solve the quadratic equation

After simplifying the polynomial by dividing it by \(x + 2\), the resulting quadratic equation can be solved by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2}-4ac} }{2a}\). Apply this formula to get the roots of the quadratic equation.

Step 3: List all solutions

List all the roots of the equation. The list includes \(-2\) (the given root) and the roots of the quadratic equation derived from the polynomial division. These will be the solutions to the original equation.

Key Concepts

Polynomial Roots

Polynomial roots are the values of the variable that make the polynomial equal to zero. These roots are also called zeros of the polynomial. When we say find the roots of a polynomial, we mean finding the solutions to the equation set equal to zero. In the exercise, \(-2\) is a given root of the cubic polynomial \(2x^3 - 3x^2 - 11x + 6\). This means when \(x = -2\), the polynomial evaluates to zero. Identifying the roots is crucial in breaking down the polynomial into simpler factors. These further help solve the polynomial completely.

To discover other roots, especially if one is given, use polynomial division or factorization techniques. This process simplifies larger polynomials step-by-step until all roots are identified.

• To verify a root, substitute it back into the polynomial and check if it results in zero.
• Every polynomial degree indicates the maximum number of roots it can have.

Quadratic Equation

A quadratic equation is of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). Solving a quadratic equation often involves using the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides solutions by addressing the quadratic formula's discriminant \((b^2 - 4ac)\). The solutions can either be real or complex.

In the given exercise, once the polynomial division was performed, a quadratic expression remained. This expression can be solved using the quadratic formula to find the other roots. Here are some straightforward steps to solve it:

• Identify the coefficients \(a\), \(b\), and \(c\) from the resulting quadratic equation.
• Compute the discriminant \(b^2 - 4ac\) to determine the nature of the roots.
• Substitute \(a\), \(b\), and \(c\) into the quadratic formula to calculate the roots.

Quadratic equations are pivotal in mathematics, providing solutions to numerous practical problems.

Cubic Polynomials

Cubic polynomials are polynomials of degree three, represented in the form \(ax^3 + bx^2 + cx + d\), where \(a eq 0\). They are integral in understanding complex mathematical concepts. The exercise deals with solving a cubic polynomial \(2x^3 - 3x^2 - 11x + 6\). Solving such polynomials often involves a mix of techniques, including synthetic division or polynomial division.

Cubic polynomials can have up to three roots, which can be a mix of real and complex numbers. To tackle these polynomials:

• Check for possible rational roots using the Rational Root Theorem.
• If one root (\(-2\) in the exercise's case) is identified, perform polynomial division to reduce it to a quadratic polynomial.
• Use the quadratic formula or factorization to solve the remaining polynomial.

Understanding cubic polynomials not only aids in solving equations but also in grasping higher numerical concepts. They appear in real-world scenarios like physics, engineering, and economics, modeling natural cubic phenomena.