Use A Graphing Calculator And A System Of Equations To Find The Roots Of The Equation: X 4 − 4 X 3 = 6 X 2 − 12 X X^4 - 4x^3 = 6x^2 - 12x X 4 − 4 X 3 = 6 X 2 − 12 X From Least To Greatest, What Are The Integral Roots Of The Equation? □ \square □ And □ \square □

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Introduction

In this article, we will explore the use of a graphing calculator and a system of equations to find the roots of the given equation: x4−4x3=6x2−12xx^4 - 4x^3 = 6x^2 - 12x. We will also determine the integral roots of the equation from least to greatest.

Understanding the Equation

The given equation is a quartic equation, which can be written as:

x4−4x3−6x2+12x=0x^4 - 4x^3 - 6x^2 + 12x = 0

This equation can be factored as:

x(x3−4x2−6x+12)=0x(x^3 - 4x^2 - 6x + 12) = 0

Factoring the Cubic Equation

To factor the cubic equation x3−4x2−6x+12x^3 - 4x^2 - 6x + 12, we can use the rational root theorem to find the possible rational roots. The rational root theorem states that if a rational number p/qp/q is a root of the polynomial, then pp must be a factor of the constant term and qq must be a factor of the leading coefficient.

In this case, the constant term is 12 and the leading coefficient is 1. Therefore, the possible rational roots are ±1,±2,±3,±4,±6,±12\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12.

Using a Graphing Calculator to Find the Roots

To find the roots of the equation, we can use a graphing calculator. We can graph the function y=x3−4x2−6x+12y = x^3 - 4x^2 - 6x + 12 and find the x-intercepts.

Using a graphing calculator, we can find that the x-intercepts are approximately x=−2,x=3,x=2x = -2, x = 3, x = 2.

Finding the Integral Roots

The integral roots of the equation are the roots that are integers. In this case, the integral roots are x=−2,x=2x = -2, x = 2.

Ordering the Integral Roots

To order the integral roots from least to greatest, we can simply arrange them in order. Therefore, the integral roots of the equation are:

x=−2,x=2x = -2, x = 2

Conclusion

In this article, we used a graphing calculator and a system of equations to find the roots of the given equation. We also determined the integral roots of the equation from least to greatest. The integral roots of the equation are x=−2,x=2x = -2, x = 2.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

  1. Factor the given equation as x(x3−4x2−6x+12)=0x(x^3 - 4x^2 - 6x + 12) = 0.
  2. Factor the cubic equation x3−4x2−6x+12x^3 - 4x^2 - 6x + 12 using the rational root theorem.
  3. Use a graphing calculator to graph the function y=x3−4x2−6x+12y = x^3 - 4x^2 - 6x + 12 and find the x-intercepts.
  4. Find the integral roots of the equation by selecting the roots that are integers.
  5. Order the integral roots from least to greatest.

Graphing Calculator Instructions

Here are the instructions for using a graphing calculator to find the roots of the equation:

  1. Enter the function y=x3−4x2−6x+12y = x^3 - 4x^2 - 6x + 12 into the graphing calculator.
  2. Graph the function and find the x-intercepts.
  3. Use the x-intercepts to find the roots of the equation.

System of Equations Approach

Here is an alternative approach to solving the equation using a system of equations:

  1. Write the given equation as a system of equations: x4−4x3=6x2−12xx^4 - 4x^3 = 6x^2 - 12x.
  2. Solve the system of equations using substitution or elimination.
  3. Find the roots of the equation by solving for x.

Conclusion

In this article, we used a graphing calculator and a system of equations to find the roots of the given equation. We also determined the integral roots of the equation from least to greatest. The integral roots of the equation are x=−2,x=2x = -2, x = 2.

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Q: What is the main goal of solving the equation x4−4x3=6x2−12xx^4 - 4x^3 = 6x^2 - 12x?

A: The main goal of solving the equation is to find the roots of the equation, which are the values of x that make the equation true.

Q: What is the difference between a graphing calculator and a system of equations approach?

A: A graphing calculator approach involves using a graphing calculator to graph the function and find the x-intercepts, while a system of equations approach involves solving the system of equations using substitution or elimination.

Q: How do I factor the cubic equation x3−4x2−6x+12x^3 - 4x^2 - 6x + 12?

A: To factor the cubic equation, you can use the rational root theorem to find the possible rational roots, and then use synthetic division or long division to divide the polynomial by the possible roots.

Q: What are the integral roots of the equation?

A: The integral roots of the equation are the roots that are integers. In this case, the integral roots are x=−2,x=2x = -2, x = 2.

Q: How do I order the integral roots from least to greatest?

A: To order the integral roots from least to greatest, you can simply arrange them in order. Therefore, the integral roots of the equation are x=−2,x=2x = -2, x = 2.

Q: Can I use a graphing calculator to find the roots of the equation?

A: Yes, you can use a graphing calculator to find the roots of the equation. Simply enter the function into the graphing calculator, graph the function, and find the x-intercepts.

Q: What is the significance of the rational root theorem?

A: The rational root theorem is a theorem that states that if a rational number p/qp/q is a root of the polynomial, then pp must be a factor of the constant term and qq must be a factor of the leading coefficient.

Q: How do I find the x-intercepts of the function y=x3−4x2−6x+12y = x^3 - 4x^2 - 6x + 12?

A: To find the x-intercepts of the function, you can use a graphing calculator to graph the function and find the x-intercepts.

Q: Can I use a system of equations approach to solve the equation?

A: Yes, you can use a system of equations approach to solve the equation. Simply write the given equation as a system of equations and solve the system using substitution or elimination.

Q: What are the benefits of using a graphing calculator to solve the equation?

A: The benefits of using a graphing calculator to solve the equation include being able to visualize the function and find the x-intercepts, and being able to solve the equation quickly and easily.

Q: Can I use a graphing calculator to find the integral roots of the equation?

A: Yes, you can use a graphing calculator to find the integral roots of the equation. Simply enter the function into the graphing calculator, graph the function, and find the x-intercepts.

Q: What is the final answer to the problem?

A: The final answer to the problem is that the integral roots of the equation are x=−2,x=2x = -2, x = 2.

Conclusion

In this article, we have answered some of the most frequently asked questions about solving the equation x4−4x3=6x2−12xx^4 - 4x^3 = 6x^2 - 12x. We have discussed the main goal of solving the equation, the difference between a graphing calculator and a system of equations approach, and how to factor the cubic equation. We have also answered questions about the integral roots of the equation, how to order the integral roots from least to greatest, and how to use a graphing calculator to find the roots of the equation.