Use Synthetic Division To Show That The Number Given To The Right Of The Equation Is A Solution Of The Equation, Then Solve The Polynomial Equation.${ 2x^3 - 11x^2 + 7x + 20 = 0; , 4 }$A. { \left{-\frac{5}{2}, -1, 4\right}$}$B.

Introduction

Synthetic division is a method used to divide polynomials by linear factors. It is a powerful tool for solving polynomial equations, as it allows us to quickly and easily determine if a given number is a solution to the equation. In this article, we will use synthetic division to show that the number given to the right of the equation is a solution of the equation, and then solve the polynomial equation.

What is Synthetic Division?

Synthetic division is a method of dividing polynomials by linear factors. It is similar to long division, but it is much faster and easier to use. The process involves dividing the polynomial by a linear factor, and then using the result to determine if the number is a solution to the equation.

How to Use Synthetic Division

To use synthetic division, we need to follow these steps:

1. Write down the coefficients of the polynomial, starting with the coefficient of the highest degree term.
2. Write down the number that we are dividing by, which is the linear factor.
3. Bring down the first coefficient.
4. Multiply the number by the first coefficient, and add the result to the second coefficient.
5. Repeat step 4 until we have multiplied the number by all of the coefficients.
6. The final result is the quotient, which is the polynomial divided by the linear factor.

Example: Using Synthetic Division to Show that a Number is a Solution

Let's use synthetic division to show that the number 4 is a solution to the equation $2x^3 - 11x^2 + 7x + 20 = 0$. We will divide the polynomial by the linear factor $(x - 4)$.

| 2 | -11 | 7 | 20 |
| --- | --- | --- | --- |
| 4 |  |  |  |
|  | 2 | -3 | 8 |
|  |  | 2 | 12 |
|  |  |  | 0 |

In this example, we can see that the final result is 0, which means that the number 4 is a solution to the equation.

Solving the Polynomial Equation

Now that we have shown that the number 4 is a solution to the equation, we can use synthetic division to solve the polynomial equation. We will divide the polynomial by the linear factor $(x - 4)$.

| 2 | -11 | 7 | 20 |
| --- | --- | --- | --- |
| 4 |  |  |  |
|  | 2 | -3 | 8 |
|  |  | 2 | 12 |
|  |  |  | 0 |

The final result is the quotient, which is the polynomial divided by the linear factor. In this case, the quotient is $2x^2 - 15x + 12$.

Finding the Other Solutions

To find the other solutions to the equation, we need to factor the quotient. We can use the quadratic formula to factor the quotient.

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, the coefficients of the quotient are $a = 2$, $b = -15$, and $c = 12$. Plugging these values into the quadratic formula, we get:

x = (15 ± √((-15)^2 - 4(2)(12))) / 2(2)
x = (15 ± √(225 - 96)) / 4
x = (15 ± √129) / 4

Simplifying the expression, we get:

x = (15 ± √129) / 4
x = (15 ± 11.36) / 4

Therefore, the other solutions to the equation are:

x = (15 + 11.36) / 4
x = 26.36 / 4
x = 6.59
x = (15 - 11.36) / 4
x = 3.64 / 4
x = -0.91

Conclusion

In this article, we used synthetic division to show that the number 4 is a solution to the equation $2x^3 - 11x^2 + 7x + 20 = 0$. We then used synthetic division to solve the polynomial equation, and found the other solutions to be $x = -0.91$ and $x = 6.59$. Synthetic division is a powerful tool for solving polynomial equations, and it can be used to quickly and easily determine if a given number is a solution to the equation.

Final Answer

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Q&A: Synthetic Division

Q: What is synthetic division?

A: Synthetic division is a method used to divide polynomials by linear factors. It is a powerful tool for solving polynomial equations, as it allows us to quickly and easily determine if a given number is a solution to the equation.

Q: How does synthetic division work?

A: To use synthetic division, we need to follow these steps:

1. Write down the coefficients of the polynomial, starting with the coefficient of the highest degree term.
2. Write down the number that we are dividing by, which is the linear factor.
3. Bring down the first coefficient.
4. Multiply the number by the first coefficient, and add the result to the second coefficient.
5. Repeat step 4 until we have multiplied the number by all of the coefficients.
6. The final result is the quotient, which is the polynomial divided by the linear factor.

Q: What is the purpose of synthetic division?

A: The purpose of synthetic division is to quickly and easily determine if a given number is a solution to a polynomial equation. It can also be used to solve polynomial equations by dividing the polynomial by a linear factor.

Q: How do I know if a number is a solution to a polynomial equation using synthetic division?

A: If the final result of the synthetic division is 0, then the number is a solution to the polynomial equation.

Q: Can I use synthetic division to solve polynomial equations with more than one linear factor?

A: Yes, you can use synthetic division to solve polynomial equations with more than one linear factor. However, you will need to repeat the process for each linear factor.

Q: What are some common mistakes to avoid when using synthetic division?

A: Some common mistakes to avoid when using synthetic division include:

• Not writing down the coefficients of the polynomial in the correct order
• Not bringing down the first coefficient
• Not multiplying the number by the correct coefficient
• Not adding the result to the correct coefficient

Q: How do I choose the number to divide by in synthetic division?

A: The number to divide by in synthetic division is the linear factor that we are testing to see if it is a solution to the polynomial equation.

Q: Can I use synthetic division to solve polynomial equations with complex coefficients?

A: Yes, you can use synthetic division to solve polynomial equations with complex coefficients. However, you will need to use complex numbers in the process.

Q: How do I know if a polynomial equation has a solution using synthetic division?

A: If the final result of the synthetic division is 0, then the polynomial equation has a solution.

Q: Can I use synthetic division to solve polynomial equations with rational coefficients?

A: Yes, you can use synthetic division to solve polynomial equations with rational coefficients. However, you will need to use rational numbers in the process.

Conclusion

Synthetic division is a powerful tool for solving polynomial equations. It allows us to quickly and easily determine if a given number is a solution to the equation, and it can be used to solve polynomial equations by dividing the polynomial by a linear factor. By following the steps outlined in this article, you can use synthetic division to solve polynomial equations and gain a deeper understanding of the subject.

Final Answer

The final answer is $\boxed{\left\{-\frac{5}{2}, -1, 4\right\}}$.