What Are The Roots Of The Polynomial Equation? 1 2 X ( X − 7 ) ( X + 9 ) = 0 \frac{1}{2} X(x-7)(x+9)=0 2 1 X ( X − 7 ) ( X + 9 ) = 0 Select Each Correct Answer.- − 9 -9 − 9 - − 7 -7 − 7 - − 1 2 -\frac{1}{2} − 2 1 - 0 0 0 - 1 2 \frac{1}{2} 2 1 - 7 7 7 - 9 9 9

Mar 13, 2025 by ADMIN 265 views

**Understanding the Roots of a Polynomial Equation**

In mathematics, a polynomial equation is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. The roots of a polynomial equation are the values of the variable that satisfy the equation, making it equal to zero. In this article, we will explore the roots of the given polynomial equation: $\frac{1}{2} x(x-7)(x+9)=0$ .

What are the Roots of a Polynomial Equation?

The roots of a polynomial equation are the values of the variable that make the equation equal to zero. In other words, if we substitute the root into the equation, the equation will be satisfied, and the result will be zero. To find the roots of a polynomial equation, we need to solve the equation for the variable.

Solving the Polynomial Equation

To solve the polynomial equation $\frac{1}{2} x(x-7)(x+9)=0$ , we need to find the values of $x$ that make the equation equal to zero. We can start by factoring the equation:

\frac{1}{2} x(x-7)(x+9)=0

x(x-7)(x+9)=0

Now, we can see that the equation is equal to zero when any of the factors are equal to zero. This means that we can set each factor equal to zero and solve for $x$ .

Finding the Roots

Let's set each factor equal to zero and solve for $x$ :

$x=0$

x(x-7)(x+9)=0

0(0-7)(0+9)=0

0(-7)(9)=0

0=0

This is true, so $x=0$ is a root of the equation.

$x-7=0$

x-7=0

x=7

This is true, so $x=7$ is a root of the equation.

$x+9=0$

x+9=0

x=-9

This is true, so $x=-9$ is a root of the equation.

Conclusion

In conclusion, the roots of the polynomial equation $\frac{1}{2} x(x-7)(x+9)=0$ are $x=0$ , $x=7$ , and $x=-9$ . These values of $x$ make the equation equal to zero, satisfying the equation.

Selecting the Correct Answer

Based on our analysis, the correct answers are:

$-9$
$-7$
$0$
$7$
$9$

Note that the value $-\frac{1}{2}$ and $\frac{1}{2}$ are not roots of the equation, as they do not make the equation equal to zero.

Final Thoughts

In our previous article, we explored the roots of a polynomial equation and how to find them. However, we understand that there may be more questions and concerns about polynomial equations. In this article, we will address some of the most frequently asked questions about polynomial equations.

Q: What is a polynomial equation?

A: A polynomial equation is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. It is a mathematical expression that can be written in the form of $ax^n + bx^{n-1} + \ldots + cx + d = 0$ , where $a$ , $b$ , $c$ , and $d$ are constants, and $x$ is the variable.

Q: What are the roots of a polynomial equation?

A: The roots of a polynomial equation are the values of the variable that make the equation equal to zero. In other words, if we substitute the root into the equation, the equation will be satisfied, and the result will be zero.

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

A: To find the roots of a polynomial equation, you need to solve the equation for the variable. This can be done by factoring the equation, using the quadratic formula, or other methods.

Q: What is the difference between a linear equation and a polynomial equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, $2x + 3 = 0$ is a linear equation. A polynomial equation, on the other hand, is an equation in which the highest power of the variable is greater than 1. For example, $x^2 + 2x + 1 = 0$ is a polynomial equation.

Q: Can a polynomial equation have more than one root?

A: Yes, a polynomial equation can have more than one root. In fact, a polynomial equation can have any number of roots, including zero, one, or multiple roots.

Q: How do I determine the number of roots of a polynomial equation?

A: To determine the number of roots of a polynomial equation, you can use the following methods:

Factor the equation: If the equation can be factored into linear factors, then the number of roots is equal to the number of factors.
Use the quadratic formula: If the equation is a quadratic equation, then the number of roots is equal to the number of solutions to the quadratic formula.
Use the Descartes' rule of signs: This rule states that the number of positive roots of a polynomial equation is equal to the number of sign changes in the coefficients of the equation, or less than that by a positive even integer.

Q: Can a polynomial equation have complex roots?

A: Yes, a polynomial equation can have complex roots. In fact, complex roots are a common occurrence in polynomial equations.

Q: How do I find the complex roots of a polynomial equation?

A: To find the complex roots of a polynomial equation, you can use the following methods:

Use the quadratic formula: If the equation is a quadratic equation, then the complex roots can be found using the quadratic formula.
Use the factoring method: If the equation can be factored into linear factors, then the complex roots can be found by solving for the variable.
Use numerical methods: If the equation is too complex to solve analytically, then numerical methods such as the Newton-Raphson method can be used to find the complex roots.

Conclusion

In this article, we have addressed some of the most frequently asked questions about polynomial equations. We have covered topics such as the definition of a polynomial equation, the roots of a polynomial equation, and how to find the roots of a polynomial equation. We have also discussed the difference between a linear equation and a polynomial equation, and how to determine the number of roots of a polynomial equation. Finally, we have covered the topic of complex roots and how to find them.