Identify The Real Roots Of The Function Below, Then Use Those Real Roots To Fill In The Sign Table.${ F(x) = -(x-5)\left(x 2-4\right)\left(x 2+4\right) }$Identify All Real Roots. Use Commas To Separate: ${ 5, 2, -2 }$Click An

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Introduction

In mathematics, solving polynomial equations is a crucial aspect of algebra. One of the key steps in solving these equations is identifying the real roots, which are the values of x that make the equation equal to zero. In this article, we will focus on identifying the real roots of a given function and then use those real roots to fill in the sign table.

The Function

The given function is:

f(x)=−(x−5)(x2−4)(x2+4){ f(x) = -(x-5)\left(x^2-4\right)\left(x^2+4\right) }

To identify the real roots, we need to set the function equal to zero and solve for x.

Identifying Real Roots

To find the real roots, we need to set each factor of the function equal to zero and solve for x.

  1. Factor 1: (x-5)

Setting (x-5) equal to zero, we get:

x−5=0{ x - 5 = 0 }

Solving for x, we get:

x=5{ x = 5 }

So, the first real root is x = 5.

  1. Factor 2: (x^2-4)

Setting (x^2-4) equal to zero, we get:

x2−4=0{ x^2 - 4 = 0 }

Solving for x, we get:

x2=4{ x^2 = 4 }

Taking the square root of both sides, we get:

x=±2{ x = \pm 2 }

So, the second and third real roots are x = 2 and x = -2.

The Real Roots

The real roots of the function are:

5,2,−2{ 5, 2, -2 }

Filling in the Sign Table

Now that we have identified the real roots, we can fill in the sign table. The sign table is a table that shows the sign of the function in different intervals.

Interval Sign of (x-5) Sign of (x^2-4) Sign of (x^2+4) Sign of f(x)
(-∞, -2) - - + +
(-2, 2) - + + -
(2, 5) - + + -
(5, ∞) + + + +

Conclusion

In this article, we identified the real roots of the given function and then used those real roots to fill in the sign table. The real roots are x = 5, x = 2, and x = -2. The sign table shows the sign of the function in different intervals, which can be used to determine the behavior of the function.

Real-World Applications

Solving polynomial equations and identifying real roots have many real-world applications. For example, in physics, the motion of an object can be modeled using polynomial equations. In engineering, polynomial equations are used to design and optimize systems. In economics, polynomial equations are used to model economic systems and make predictions.

Tips and Tricks

When solving polynomial equations, it's essential to identify the real roots first. This can be done by setting each factor of the function equal to zero and solving for x. Once the real roots are identified, the sign table can be filled in to determine the behavior of the function.

Common Mistakes

One common mistake when solving polynomial equations is to forget to check for real roots. It's essential to check for real roots first, as they can affect the behavior of the function.

Final Thoughts

Introduction

In our previous article, we discussed how to solve polynomial equations and identify real roots. In this article, we will answer some frequently asked questions related to solving polynomial equations and identifying real roots.

Q: What is a polynomial equation?

A polynomial equation is an equation in which the highest power of the variable (usually x) is a non-negative integer. For example, 2x^2 + 3x - 4 is a polynomial equation.

A: A polynomial equation is an equation in which the highest power of the variable (usually x) is a non-negative integer.

Q: How do I identify the real roots of a polynomial equation?

To identify the real roots of a polynomial equation, you need to set each factor of the equation equal to zero and solve for x.

A: To identify the real roots of a polynomial equation, you need to set each factor of the equation equal to zero and solve for x.

Q: What is a factor of a polynomial equation?

A factor of a polynomial equation is an expression that divides the polynomial equation exactly. For example, (x-2) is a factor of the polynomial equation x^2 - 4x + 4.

A: A factor of a polynomial equation is an expression that divides the polynomial equation exactly.

Q: How do I fill in the sign table?

To fill in the sign table, you need to determine the sign of the function in different intervals. You can do this by using the real roots of the function and the sign of each factor.

A: To fill in the sign table, you need to determine the sign of the function in different intervals. You can do this by using the real roots of the function and the sign of each factor.

Q: What is the significance of the sign table?

The sign table is a table that shows the sign of the function in different intervals. It can be used to determine the behavior of the function and to identify the intervals where the function is positive or negative.

A: The sign table is a table that shows the sign of the function in different intervals. It can be used to determine the behavior of the function and to identify the intervals where the function is positive or negative.

Q: How do I use the sign table to determine the behavior of the function?

To use the sign table to determine the behavior of the function, you need to look at the sign of the function in each interval. If the sign is positive, the function is increasing. If the sign is negative, the function is decreasing.

A: To use the sign table to determine the behavior of the function, you need to look at the sign of the function in each interval. If the sign is positive, the function is increasing. If the sign is negative, the function is decreasing.

Q: What are some common mistakes to avoid when solving polynomial equations?

Some common mistakes to avoid when solving polynomial equations include:

  • Forgetting to check for real roots
  • Not setting each factor of the equation equal to zero
  • Not using the sign table to determine the behavior of the function

A: Some common mistakes to avoid when solving polynomial equations include:

  • Forgetting to check for real roots
  • Not setting each factor of the equation equal to zero
  • Not using the sign table to determine the behavior of the function

Conclusion

In this article, we answered some frequently asked questions related to solving polynomial equations and identifying real roots. We hope that this article has been helpful in clarifying some of the concepts related to solving polynomial equations.

Real-World Applications

Solving polynomial equations and identifying real roots have many real-world applications. For example, in physics, the motion of an object can be modeled using polynomial equations. In engineering, polynomial equations are used to design and optimize systems. In economics, polynomial equations are used to model economic systems and make predictions.

Tips and Tricks

When solving polynomial equations, it's essential to identify the real roots first. This can be done by setting each factor of the equation equal to zero and solving for x. Once the real roots are identified, the sign table can be filled in to determine the behavior of the function.

Common Mistakes

One common mistake when solving polynomial equations is to forget to check for real roots. It's essential to check for real roots first, as they can affect the behavior of the function.

Final Thoughts

Solving polynomial equations and identifying real roots is a crucial aspect of algebra. By following the steps outlined in this article, you can solve polynomial equations and identify real roots. Remember to check for real roots first and use the sign table to determine the behavior of the function.