Factoring Polynomials: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of polynomial factoring, a crucial skill in algebra. We'll tackle a specific problem: finding the completely factored form of the cubic polynomial f(x) = 6x³ - 13x² - 4x + 15. Don't worry if this seems daunting; we'll break it down step-by-step, making it easy to understand. Factoring is like reverse distribution; we're essentially trying to rewrite the polynomial as a product of simpler expressions. This is super useful for solving equations, simplifying expressions, and understanding the behavior of functions. Are you ready to get started? Let's jump in!

Understanding the Problem and Key Concepts

Before we start, let's make sure we're all on the same page. The goal is to rewrite the polynomial as a product of its factors. Remember that a factor is an expression that divides evenly into another expression. When we say "completely factored," it means we've broken down the polynomial into its simplest factors, which are usually linear (like x + a) or irreducible quadratic expressions. For our polynomial, 6x³ - 13x² - 4x + 15, we are looking for linear factors. This means, we want to find expressions in the form of (ax + b) that, when multiplied together, give us our original polynomial. One key concept is the Factor Theorem. This theorem tells us that if a polynomial f(x) has a factor (x - k), then f(k) = 0. This is our secret weapon for finding the roots or zeroes of the polynomial, which will, in turn, help us find the factors. We'll use this later. Another thing to keep in mind is the Rational Root Theorem, which helps us narrow down the possible rational roots of a polynomial. This theorem states that any rational root must be a factor of the constant term divided by a factor of the leading coefficient. So, in our case, we're looking for factors of 15 divided by factors of 6. This will give us a list of possible rational roots to test.

Step-by-Step Factoring Process

Alright, let's start factoring! We will start by using the Rational Root Theorem to find the possible rational roots. Then, we'll test these possible roots using synthetic division (or polynomial long division) to see if they are actually roots of the polynomial. Once we find a root, we know that (x - root) is a factor. Finally, we will factorize the resulting quotient by using different techniques. Let's get started!

Identifying Possible Rational Roots

As mentioned, the Rational Root Theorem is our first tool. The factors of the constant term, 15, are ±1, ±3, ±5, and ±15. The factors of the leading coefficient, 6, are ±1, ±2, ±3, and ±6. Therefore, the possible rational roots are: ±1, ±3, ±5, ±15, ±1/2, ±3/2, ±5/2, and ±15/2. This might seem like a lot, but don't worry. We only need to find one root, and we can use it to reduce the degree of the polynomial and find the other roots. Now, let's test these values!

Testing Possible Roots Using Synthetic Division

We can use synthetic division to quickly test whether these potential roots are actual roots. Synthetic division is a streamlined method of polynomial division. Remember that if the remainder is 0, then the tested value is a root. Let's start by testing x = 1. We will arrange our coefficients: 6, -13, -4, 15 and proceed with synthetic division.

The remainder is 4, so x = 1 is not a root. Let's try x = -1.

Excellent! The remainder is 0, which means x = -1 is a root! This also means that (x + 1) is a factor of our polynomial. The result of the synthetic division gives us the coefficients of the remaining quadratic expression. So we now have (x + 1)(6x² - 19x + 15).

Factoring the Quadratic Expression

Now, we have to factor the quadratic expression 6x² - 19x + 15. We can do this by factoring by grouping, or by using the quadratic formula, but we can find the roots of the quadratic formula or any method to factor the quadratic expression, so we get the expression:

6x² - 19x + 15 = (2x - 3)(3x - 5).

Thus, our completely factored polynomial is:

(x + 1)(2x - 3)(3x - 5).

The Final Answer and Explanation

Therefore, the completely factored form of f(x) = 6x³ - 13x² - 4x + 15 is (x + 1)(2x - 3)(3x - 5). To get this answer, we used the Rational Root Theorem to find possible rational roots, used synthetic division to find one root, and then factored the resulting quadratic expression. Factoring is a fundamental concept in algebra, and this process will help you solve equations, simplify expressions, and understand the behavior of functions. Great job!

Summary and Additional Tips

In summary, we learned how to factor a cubic polynomial by:

1. Identifying possible rational roots using the Rational Root Theorem.
2. Testing the roots using synthetic division.
3. Factoring the resulting quadratic expression (or continuing synthetic division if the degree is higher than 2).

Here are some additional tips:

• Practice, practice, practice: The more you factor, the better you'll become at recognizing patterns and finding factors. There are several online resources and textbooks with factoring exercises.
• Check your work: Always multiply your factors back out to make sure you get the original polynomial. This is an easy way to catch errors.
• Look for common factors: Always check if there is a common factor that you can take out first. This can simplify the factoring process.
• Familiarize yourself with different factoring techniques: Besides factoring by grouping and using the quadratic formula, understand other techniques like difference of squares, sum/difference of cubes, and perfect square trinomials.

With these skills and tips, you will be factoring polynomials like a pro in no time. Keep up the great work, and always remember that every step you take strengthens your mathematical foundation!