What Are The Roots Of The Polynomial Equation $x^3 - 2x^2 - 5x + 5$? Use A Graphing Calculator And A System Of Equations To Find Non-integer Roots To The Nearest Hundredth.A. $-3, -5$B. $-2.24, -2, 2.24$C. $-2.24, 2,

Introduction

In mathematics, polynomial equations are a fundamental concept in algebra, and finding their roots is a crucial aspect of solving these equations. The roots of a polynomial equation are the values of the variable that satisfy the equation, and they can be real or complex numbers. In this article, we will explore the roots of the polynomial equation $x^3 - 2x^2 - 5x + 5$ using a graphing calculator and a system of equations.

Using a Graphing Calculator

A graphing calculator is a powerful tool that can help us visualize the graph of a polynomial equation and find its roots. To use a graphing calculator, we need to enter the equation $x^3 - 2x^2 - 5x + 5$ and set the calculator to graph the equation. The graph will show us the behavior of the equation and help us identify the roots.

Graphing the Equation

To graph the equation $x^3 - 2x^2 - 5x + 5$, we need to enter the equation into the graphing calculator. We can do this by pressing the "Y=" button and entering the equation. The calculator will then graph the equation and show us the behavior of the equation.

Finding the Roots

Once we have graphed the equation, we can use the calculator to find the roots of the equation. We can do this by pressing the "2nd" button and selecting the "zero" option. The calculator will then show us the roots of the equation.

Using a System of Equations

Another way to find the roots of the polynomial equation $x^3 - 2x^2 - 5x + 5$ is to use a system of equations. We can do this by setting the equation equal to zero and solving for x.

Setting the Equation Equal to Zero

To set the equation equal to zero, we need to subtract the constant term from both sides of the equation. This gives us the equation:

$x^3 - 2x^2 - 5x = -5$

Solving for x

To solve for x, we can use the quadratic formula. The quadratic formula is a mathematical formula that gives us the solutions to a quadratic equation. In this case, we can use the quadratic formula to solve for x.

The Quadratic Formula

The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this case, we have:

$a = 1$ $b = -2$ $c = -5$

Substituting the Values

Substituting the values of a, b, and c into the quadratic formula, we get:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-5)}}{2(1)}$

Simplifying the expression, we get:

$x = \frac{2 \pm \sqrt{4 + 20}}{2}$

$x = \frac{2 \pm \sqrt{24}}{2}$

$x = \frac{2 \pm 2\sqrt{6}}{2}$

$x = 1 \pm \sqrt{6}$

Finding the Non-Integer Roots

To find the non-integer roots of the polynomial equation $x^3 - 2x^2 - 5x + 5$, we need to use the quadratic formula to solve for x. We can do this by substituting the values of a, b, and c into the quadratic formula and simplifying the expression.

Simplifying the Expression

Simplifying the expression, we get:

$x = 1 \pm \sqrt{6}$

$x = 1 \pm 2.45$

$x = 3.45$ or $x = -1.45$

Conclusion

In conclusion, we have used a graphing calculator and a system of equations to find the roots of the polynomial equation $x^3 - 2x^2 - 5x + 5$. We have found that the roots of the equation are $-3, -5$ and $-2.24, 2.24$. These roots are non-integer values that satisfy the equation.

Discussion

The roots of a polynomial equation are a fundamental concept in algebra, and finding them is a crucial aspect of solving these equations. In this article, we have used a graphing calculator and a system of equations to find the roots of the polynomial equation $x^3 - 2x^2 - 5x + 5$. We have found that the roots of the equation are $-3, -5$ and $-2.24, 2.24$. These roots are non-integer values that satisfy the equation.

Final Answer

The final answer is $-2.24, 2.24$.