Solve The Polynomial Equation By Applying The Zero Product Property.${ 21r^2 - 14r = 0 }$A. { R = -7 $}$ Or { R = \frac{2}{3} $}$ B. { R = 0 $}$ Or { R = -\frac{3}{2} $}$ C. [$ R = 0

Introduction

In algebra, the zero product property is a fundamental concept used to solve polynomial equations. It states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In this article, we will apply the zero product property to solve a given polynomial equation.

Understanding the Zero Product Property

The zero product property is a simple yet powerful concept that can be used to solve a wide range of polynomial equations. It is based on the idea that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. This property can be expressed mathematically as:

• If $ab = 0$, then $a = 0$ or $b = 0$
• If $abc = 0$, then $a = 0$, $b = 0$, or $c = 0$
• If $abcd = 0$, then $a = 0$, $b = 0$, $c = 0$, or $d = 0$

Applying the Zero Product Property to the Given Polynomial Equation

The given polynomial equation is:

${ 21r^2 - 14r = 0 }$

To apply the zero product property, we need to factor the left-hand side of the equation. We can do this by factoring out the greatest common factor (GCF) of the two terms.

The GCF of $21r^2$ and $-14r$ is $7r$. Therefore, we can factor the left-hand side of the equation as follows:

${ 21r^2 - 14r = 7r(3r - 2) = 0 }$

Now, we can apply the zero product property to solve for $r$. We have two factors: $7r$ and $(3r - 2)$. Since the product of these two factors is equal to zero, we know that at least one of them must be equal to zero.

Solving for $r$

We can set each factor equal to zero and solve for $r$.

Solving for $7r = 0$

If $7r = 0$, then we can divide both sides of the equation by 7 to get:

${ r = 0 }$

Solving for $(3r - 2) = 0$

If $(3r - 2) = 0$, then we can add 2 to both sides of the equation to get:

${ 3r = 2 }$

Next, we can divide both sides of the equation by 3 to get:

${ r = \frac{2}{3} }$

Conclusion

In this article, we applied the zero product property to solve a given polynomial equation. We factored the left-hand side of the equation and then applied the zero product property to solve for $r$. We found that $r = 0$ or $r = \frac{2}{3}$.

Answer

The correct answer is:

${ r = 0 }$ or ${ r = \frac{2}{3} }$

Discussion

The zero product property is a fundamental concept in algebra that can be used to solve a wide range of polynomial equations. In this article, we applied the zero product property to solve a given polynomial equation and found that $r = 0$ or $r = \frac{2}{3}$.

Example Problems

Here are some example problems that you can try to practice applying the zero product property:

1. Solve the polynomial equation $x^2 + 5x = 0$ using the zero product property.
2. Solve the polynomial equation $2x^2 - 3x = 0$ using the zero product property.
3. Solve the polynomial equation $x^2 + 2x - 3 = 0$ using the zero product property.

Tips and Tricks

Here are some tips and tricks that you can use to apply the zero product property:

• Make sure to factor the left-hand side of the equation before applying the zero product property.
• Use the zero product property to solve for each factor separately.
• Check your work by plugging your solutions back into the original equation.

Conclusion

Introduction

In our previous article, we applied the zero product property to solve a given polynomial equation. In this article, we will answer some frequently asked questions (FAQs) about the zero product property and how to apply it to solve polynomial equations.

Q&A

Q: What is the zero product property?

A: The zero product property is a fundamental concept in algebra that states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero.

Q: How do I apply the zero product property to solve a polynomial equation?

A: To apply the zero product property, you need to factor the left-hand side of the equation and then set each factor equal to zero. You can then solve for the variable and find the solutions to the equation.

Q: What are some common mistakes to avoid when applying the zero product property?

A: Some common mistakes to avoid when applying the zero product property include:

• Not factoring the left-hand side of the equation before applying the zero product property.
• Not setting each factor equal to zero and solving for the variable.
• Not checking your work by plugging your solutions back into the original equation.

Q: Can I use the zero product property to solve polynomial equations with more than two factors?

A: Yes, you can use the zero product property to solve polynomial equations with more than two factors. Simply factor the left-hand side of the equation and then set each factor equal to zero. You can then solve for the variable and find the solutions to the equation.

Q: How do I know if I have factored the left-hand side of the equation correctly?

A: To check if you have factored the left-hand side of the equation correctly, you can use the following steps:

• Check if the factors are correct by multiplying them together and making sure they equal the original equation.
• Check if the factors are in the correct order by rearranging them and making sure they still equal the original equation.

Q: Can I use the zero product property to solve polynomial equations with fractions?

A: Yes, you can use the zero product property to solve polynomial equations with fractions. Simply factor the left-hand side of the equation and then set each factor equal to zero. You can then solve for the variable and find the solutions to the equation.

Q: How do I check my work when applying the zero product property?

A: To check your work when applying the zero product property, you can use the following steps:

• Plug your solutions back into the original equation and make sure they are true.
• Check if the factors are correct by multiplying them together and making sure they equal the original equation.
• Check if the factors are in the correct order by rearranging them and making sure they still equal the original equation.

Example Problems

Here are some example problems that you can try to practice applying the zero product property:

1. Solve the polynomial equation $x^2 + 5x = 0$ using the zero product property.
2. Solve the polynomial equation $2x^2 - 3x = 0$ using the zero product property.
3. Solve the polynomial equation $x^2 + 2x - 3 = 0$ using the zero product property.

Tips and Tricks

Here are some tips and tricks that you can use to apply the zero product property:

• Make sure to factor the left-hand side of the equation before applying the zero product property.
• Use the zero product property to solve for each factor separately.
• Check your work by plugging your solutions back into the original equation.

Conclusion

In conclusion, the zero product property is a fundamental concept in algebra that can be used to solve a wide range of polynomial equations. By factoring the left-hand side of the equation and then applying the zero product property, we can solve for the variable and find the solutions to the equation. Remember to check your work by plugging your solutions back into the original equation and making sure they are true.