Factoring Polynomials Using Synthetic Division A Step-by-Step Guide

Hey guys! Factoring polynomials can seem like cracking a secret code, but with the right tools, it becomes a breeze. In this article, we'll dive deep into using synthetic division, a super handy technique, to factor the polynomial $x^4 + 6x^3 + 33x^2 + 150x + 200$. We'll break down the process step-by-step, making it crystal clear how to arrive at the correct factored form. So, grab your thinking caps, and let's get started!

Understanding the Problem: What are we trying to achieve?

Before we jump into the solution, let's quickly recap what it means to factor a polynomial. Essentially, factoring is like reverse multiplication. We're trying to break down the given polynomial into a product of simpler expressions, usually binomials (expressions with two terms) or trinomials (expressions with three terms). The factored form makes it much easier to analyze the polynomial's behavior, find its roots (where the polynomial equals zero), and sketch its graph. In our specific problem, we're presented with a fourth-degree polynomial (the highest power of x is 4), which means it can have up to four roots. Our mission is to find the factors that, when multiplied together, give us the original polynomial. And we'll be using the power of synthetic division to make this happen!

Synthetic Division: Our Key Tool

Synthetic division is a streamlined method for dividing a polynomial by a linear factor of the form (x - c), where 'c' is a constant. It's a shortcut compared to long division and is particularly useful for finding roots and factoring polynomials. The beauty of synthetic division lies in its simplicity and efficiency. Instead of dealing with variables and exponents directly, we focus on the coefficients of the polynomial. This makes the process less prone to errors and significantly faster. The result of synthetic division tells us two crucial things: the quotient (the result of the division) and the remainder. If the remainder is zero, it means that (x - c) is a factor of the polynomial, and 'c' is a root. This is exactly what we need for factoring! Let's see how this works in action with our polynomial.

Step-by-Step Solution: Cracking the Code

Now, let's roll up our sleeves and factor the polynomial $x^4 + 6x^3 + 33x^2 + 150x + 200$ using synthetic division.

1. Finding Potential Roots: The Rational Root Theorem

The first step in using synthetic division effectively is to identify potential roots. The Rational Root Theorem is our trusty guide here. This theorem states that if a polynomial has integer coefficients, any rational root (a root that can be expressed as a fraction p/q) must have 'p' as a factor of the constant term and 'q' as a factor of the leading coefficient. In our case, the constant term is 200, and the leading coefficient is 1. This means our potential rational roots are the factors of 200 (both positive and negative). These factors include ±1, ±2, ±4, ±5, ±8, ±10, ±20, ±25, ±40, ±50, ±100, and ±200. That's quite a list, but don't worry, we won't have to try them all! We'll start with the smaller, simpler factors first.

2. Testing Potential Roots: Synthetic Division in Action

Let's start by testing -2 as a potential root. We set up our synthetic division table as follows:

-2 | 1 6 33 150 200
   |________________________

• Bring down the leading coefficient (1) to the bottom row.

-2 | 1 6 33 150 200
   |________________________
   1

• Multiply -2 by the number we just brought down (1), and write the result (-2) under the next coefficient (6).

-2 | 1 6 33 150 200
   | -2
   |________________________
   1

• Add the numbers in the second column (6 and -2) and write the sum (4) in the bottom row.

-2 | 1 6 33 150 200
   | -2
   |________________________
   1 4

• Repeat the process: Multiply -2 by 4, write the result (-8) under 33, add 33 and -8, and write the sum (25) in the bottom row.

-2 | 1 6 33 150 200
   | -2 -8
   |________________________
   1 4 25

• Continue this process until we reach the last column.

-2 | 1 6 33 150 200
   | -2 -8 -50 -200
   |________________________
   1 4 25 100 0

The last number in the bottom row is the remainder. In this case, the remainder is 0. This is fantastic news! It means that (x + 2) is a factor of our polynomial, and -2 is a root.

3. The Quotient Polynomial

The other numbers in the bottom row (1, 4, 25, and 100) represent the coefficients of the quotient polynomial. Since we divided a fourth-degree polynomial by a linear factor (x + 2), the quotient will be a third-degree polynomial. So, the quotient is $x^3 + 4x^2 + 25x + 100$. Now we have:

$$x^4 + 6x^3 + 33x^2 + 150x + 200 = (x + 2)(x^3 + 4x^2 + 25x + 100)$$

4. Factoring the Quotient: Rinse and Repeat

We're not done yet! We need to factor the cubic polynomial $x^3 + 4x^2 + 25x + 100$. Let's use synthetic division again. We'll try -4 as a potential root (again, using the Rational Root Theorem on the quotient). The synthetic division looks like this:

-4 | 1 4 25 100
   | -4 0 -100
   |________________
   1 0 25 0

Again, we have a remainder of 0! This means (x + 4) is a factor of the quotient polynomial. The new quotient is $x^2 + 0x + 25$, which simplifies to $x^2 + 25$. So now we have:

$$x^4 + 6x^3 + 33x^2 + 150x + 200 = (x + 2)(x + 4)(x^2 + 25)$$

5. The Final Factor: A Sum of Squares

The last factor, $x^2 + 25$, is a sum of squares. Remember, a sum of squares ($a^2 + b^2$) does not factor over real numbers. However, it can be factored over complex numbers as (x + 5i)(x - 5i), where 'i' is the imaginary unit (√-1). But since the answer choices provided only deal with real number factors, we'll leave it as $x^2 + 25$.

The Answer: Cracking the Code!

Therefore, the factored form of the polynomial $x^4 + 6x^3 + 33x^2 + 150x + 200$ is:

(A) $(x + 2)(x + 4)(x^2 + 25)$

Key Takeaways: Mastering Polynomial Factoring

• Synthetic division is your friend: It's a powerful tool for dividing polynomials and finding roots quickly.
• The Rational Root Theorem is your guide: It helps you narrow down the list of potential rational roots.
• Remainder Theorem: If the remainder after synthetic division is 0, you've found a factor!
• Don't forget the quotient: The numbers in the bottom row (excluding the remainder) give you the coefficients of the quotient polynomial.
• Keep factoring: Continue the process until you have factored the polynomial completely.

Practice Makes Perfect: Keep Honing Your Skills

Factoring polynomials might seem daunting at first, but like any skill, it gets easier with practice. The more you work with synthetic division and the Rational Root Theorem, the more comfortable you'll become. So, grab some practice problems, and keep those factoring muscles flexing! You've got this!

If you have any questions or want to explore more factoring techniques, feel free to ask. Happy factoring, everyone!