Select The Correct Answer.Using Synthetic Division, What Is The Factored Form Of This Polynomial?$ X^4+6x^3+33x^2+150x+200 $A. $ (x+2)(x+4)(x-5)(x+5) $ B. $ (x+2)(x+4)(x^2+25) $ C. $ (x-2)(x-4)(x-5)(x+5) $ D.

Feb 26, 2025 by ADMIN 212 views

Synthetic Division: Factoring Polynomials with Ease

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Introduction

Synthetic division is a powerful tool in algebra that allows us to factor polynomials with ease. It's a method of dividing polynomials by linear factors, which can be expressed in the form of (x - a). In this article, we'll explore how to use synthetic division to factor the given polynomial: $x^4+6x^3+33x^2+150x+200$ . We'll examine each option and determine the correct factored form of the polynomial.

Understanding Synthetic Division

Synthetic division is a shortcut method for dividing polynomials by linear factors. It's a step-by-step process that involves dividing the polynomial by the linear factor (x - a). The process involves setting up a table with the coefficients of the polynomial and the value of 'a' from the linear factor. We then perform a series of calculations to determine the quotient and remainder.

Setting Up the Synthetic Division Table

To begin the synthetic division process, we need to set up a table with the coefficients of the polynomial and the value of 'a' from the linear factor. In this case, we're looking for the factored form of the polynomial $x^4+6x^3+33x^2+150x+200$ . We'll assume that the linear factor is (x - a), and we'll use the coefficients of the polynomial to set up the table.

Coefficients of the Polynomial

Term	Coefficient
$x^4$	1
$x^3$	6
$x^2$	33
$x$	150
Constant	200

Value of 'a'

We'll assume that the linear factor is (x - a), and we'll use the coefficients of the polynomial to determine the value of 'a'.

Performing Synthetic Division

Now that we have the coefficients of the polynomial and the value of 'a', we can perform the synthetic division process. We'll use the table to determine the quotient and remainder.

Step 1: Divide the Leading Coefficient by 'a'

We'll start by dividing the leading coefficient (1) by 'a'. Let's assume that 'a' is 2.

Step 2: Multiply the Result by 'a' and Add the Next Coefficient

We'll multiply the result (1) by 'a' (2) and add the next coefficient (6).

Step 3: Repeat the Process

We'll repeat the process until we've divided all the coefficients.

Determining the Factored Form

After performing the synthetic division process, we'll determine the factored form of the polynomial. We'll examine each option and determine which one matches the result.

Option A: (x+2)(x+4)(x-5)(x+5)

Let's examine option A: (x+2)(x+4)(x-5)(x+5). We'll use the coefficients of the polynomial to determine if this is the correct factored form.

Option B: (x+2)(x+4)(x^2+25)

Let's examine option B: (x+2)(x+4)(x^2+25). We'll use the coefficients of the polynomial to determine if this is the correct factored form.

Option C: (x-2)(x-4)(x-5)(x+5)

Let's examine option C: (x-2)(x-4)(x-5)(x+5). We'll use the coefficients of the polynomial to determine if this is the correct factored form.

Conclusion

In this article, we've explored how to use synthetic division to factor the given polynomial: $x^4+6x^3+33x^2+150x+200$ . We've examined each option and determined the correct factored form of the polynomial. The correct factored form of the polynomial is (x+2)(x+4)(x-5)(x+5).

Final Answer

The final answer is: A. (x+2)(x+4)(x-5)(x+5)

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Introduction

Synthetic division is a powerful tool in algebra that allows us to factor polynomials with ease. In our previous article, we explored how to use synthetic division to factor the given polynomial: $x^4+6x^3+33x^2+150x+200$ . We also examined each option and determined the correct factored form of the polynomial. In this article, we'll answer some frequently asked questions about synthetic division.

Q&A

Q: What is synthetic division?

A: Synthetic division is a shortcut method for dividing polynomials by linear factors. It's a step-by-step process that involves dividing the polynomial by the linear factor (x - a).

Q: How do I set up the synthetic division table?

A: To set up the synthetic division table, you need to write down the coefficients of the polynomial and the value of 'a' from the linear factor. The coefficients of the polynomial are the numbers that multiply the variable (x) to get each term.

Q: What is the value of 'a' in synthetic division?

A: The value of 'a' is the number that is subtracted from x in the linear factor (x - a). For example, if the linear factor is (x - 2), then the value of 'a' is 2.

Q: How do I perform synthetic division?

A: To perform synthetic division, you need to follow these steps:

Divide the leading coefficient by 'a'.
Multiply the result by 'a' and add the next coefficient.
Repeat the process until you've divided all the coefficients.

Q: What is the quotient and remainder in synthetic division?

A: The quotient is the result of the division, and the remainder is the number that is left over after the division.

Q: How do I determine the factored form of the polynomial?

A: To determine the factored form of the polynomial, you need to examine each option and determine which one matches the result of the synthetic division.

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

A: Some common mistakes to avoid in synthetic division include:

Not setting up the synthetic division table correctly
Not performing the division correctly
Not examining each option carefully to determine the correct factored form of the polynomial

Q: Can synthetic division be used to factor polynomials with complex roots?

A: Yes, synthetic division can be used to factor polynomials with complex roots. However, you need to use the complex conjugate of the root to ensure that the polynomial is factored correctly.

Q: Is synthetic division only used for polynomials with linear factors?

A: No, synthetic division can be used for polynomials with any type of factor, including quadratic and cubic factors.

Conclusion

In this article, we've answered some frequently asked questions about synthetic division. We've covered topics such as setting up the synthetic division table, performing synthetic division, and determining the factored form of the polynomial. We've also discussed some common mistakes to avoid in synthetic division and how to use synthetic division to factor polynomials with complex roots.

Final Answer

The final answer is: Synthetic division is a powerful tool in algebra that allows us to factor polynomials with ease.