Factoring Polynomials: A Comprehensive Guide

Aug 7, 2025 by ADMIN 45 views

Factoring Polynomials: A Step-by-Step Guide

Understanding Linear Factors and Zeros

Hey math enthusiasts! Let's dive into the cool world of factoring polynomials. Today, we're going to tackle the problem of factoring a cubic polynomial, $f(x) = 3x^3 + 8x^2 - 123x + 40$ , into linear factors. We're given a hint: $k = 5$ is a zero of the function. But what does all this jargon mean? Let's break it down. First things first, a linear factor is simply a factor of the form $(x - a)$ , where 'a' is a constant. Think of it like this: if you can express your polynomial as a product of these linear factors, you've essentially simplified it to its bare bones. The zeros of a function are the values of x that make the function equal to zero. When we know a zero of the function, we also know that $(x - k)$ is a factor of $f(x)$ . In our case, knowing that 5 is a zero tells us that $(x - 5)$ must be a factor. It's like having a secret key that unlocks a part of the puzzle. So, our mission is to find the remaining factors and express the whole polynomial as a product of simple linear expressions. This process isn't just about getting the right answer; it's about understanding how polynomials behave, how their structure is connected to their roots, and how we can manipulate them to solve problems. We start with a cubic function because it adds a layer of complexity. It helps us understand how different degrees of polynomials and their factors and roots behave. As we go through the process, we'll see how strategic use of division and careful attention to detail can lead us to the solution.

We'll utilize the Factor Theorem, a key concept in polynomial algebra. This theorem states that if $k$ is a zero of a polynomial $f(x)$ , then $(x - k)$ is a factor of $f(x)$ . We already know that 5 is a zero, thus $(x - 5)$ must be a factor of our polynomial. Now, we need to figure out what multiplies with $(x - 5)$ to give us the original cubic polynomial. We are going to use polynomial division to determine the other factors. Remember, the ultimate goal is to rewrite our cubic polynomial $f(x)$ as a product of linear factors, which makes it easier to analyze, solve for its roots, and even sketch its graph. It’s important to remember that polynomials are everywhere. They are used in various fields, from engineering and physics to economics and computer science. Understanding how to factor polynomials is a fundamental skill that allows you to solve more complex problems. This article is structured as a practical guide, using the given values to illustrate the process, along with explanations to make it easier to follow. This ensures that the core principles are clear, and you can apply them to your own problems. Let's jump in and see how it's done!

Polynomial Division: The Heart of the Matter

Alright, let's roll up our sleeves and get to the core of the problem: polynomial division. Since we know $(x - 5)$ is a factor of $f(x) = 3x^3 + 8x^2 - 123x + 40$ , we'll divide $f(x)$ by $(x - 5)$ . This process will give us the remaining factor, a quadratic, which we can then break down further into linear factors. Think of polynomial division as a systematic way of reversing the multiplication process. We will use long division.

Here's how it unfolds:

Set up the division: Write down the polynomial $f(x)$ inside the division symbol and $(x - 5)$ outside.
```
    _________
```

x - 5 | 3x^3 + 8x^2 - 123x + 40 ```

Divide the first terms: Divide the first term of the polynomial ( $3x^3$ ) by the first term of the divisor ( $x$ ). This gives us $3x^2$ . Write this above the division symbol.
```
    3x^2     
```

x - 5 | 3x^3 + 8x^2 - 123x + 40 ```

Multiply: Multiply the divisor $(x - 5)$ by $3x^2$ . This gives us $3x^3 - 15x^2$ . Write this below the polynomial.
```
    3x^2     
```

x - 5 | 3x^3 + 8x^2 - 123x + 40 3x^3 - 15x^2 ```

Subtract: Subtract the result from the polynomial. $(3x^3 + 8x^2) - (3x^3 - 15x^2) = 23x^2$ . Bring down the next term (-123x).
```
    3x^2     
```

x - 5 | 3x^3 + 8x^2 - 123x + 40 3x^3 - 15x^2 __________ 23x^2 - 123x ```

Repeat: Divide the first term of the new polynomial ( $23x^2$ ) by $x$ . This gives us $23x$ . Write this above the division symbol.
```
    3x^2 + 23x  
```

x - 5 | 3x^3 + 8x^2 - 123x + 40 3x^3 - 15x^2 __________ 23x^2 - 123x ```

Multiply and Subtract: Multiply the divisor $(x - 5)$ by $23x$ . This gives us $23x^2 - 115x$ . Subtract this from the current polynomial.
```
    3x^2 + 23x  
```

x - 5 | 3x^3 + 8x^2 - 123x + 40 3x^3 - 15x^2 __________ 23x^2 - 123x 23x^2 - 115x __________ -8x + 40 ```

Repeat Again: Divide $-8x$ by $x$ which gives us $-8$ . Write it above.
```
    3x^2 + 23x - 8 
```

x - 5 | 3x^3 + 8x^2 - 123x + 40 3x^3 - 15x^2 __________ 23x^2 - 123x 23x^2 - 115x __________ -8x + 40 ```

Multiply and Subtract: Multiply $(x - 5)$ by $-8$ . This gives us $-8x + 40$ . Subtract this from the current polynomial. The result is 0, meaning we have no remainder.
```
    3x^2 + 23x - 8 
```

x - 5 | 3x^3 + 8x^2 - 123x + 40 3x^3 - 15x^2 __________ 23x^2 - 123x 23x^2 - 115x __________ -8x + 40 -8x + 40 __________ 0 ```

So, the result of the division is $3x^2 + 23x - 8$ . This means that $f(x) = (x - 5)(3x^2 + 23x - 8)$ .

Factoring the Quadratic: The Final Stretch

Great job, guys! We've successfully completed the polynomial division and have now reduced the cubic polynomial to the product of a linear factor and a quadratic factor. Now, it's time to factor the quadratic expression $3x^2 + 23x - 8$ . Our aim is to find two numbers that multiply to give us $(3 imes -8 = -24)$ and add up to 23. After playing around with different number combinations, the numbers are $24$ and $-1$ . We rewrite the middle term, $23x$ , using these two numbers.

3x^2 + 23x - 8 = 3x^2 + 24x - x - 8

Next, we will group the terms and factor by grouping:

3x^2 + 24x - x - 8 = 3x(x + 8) - 1(x + 8)

Now, we can factor out the common factor $(x + 8)$ :

3x(x + 8) - 1(x + 8) = (3x - 1)(x + 8)

So, the quadratic factor $3x^2 + 23x - 8$ factors into $(3x - 1)(x + 8)$ . Therefore, the fully factored form of the original cubic polynomial is:

f(x) = (x - 5)(3x - 1)(x + 8)

Conclusion: Putting It All Together

And there you have it! We've taken the cubic polynomial $f(x) = 3x^3 + 8x^2 - 123x + 40$ and successfully factored it into linear factors: $(x - 5)$ , $(3x - 1)$ , and $(x + 8)$ . We started with the information that $k = 5$ is a zero, used the Factor Theorem, and employed polynomial division to simplify the polynomial. By breaking down the cubic polynomial step-by-step, we were able to understand the process more fully. This whole process highlights the interconnectedness of zeros, factors, and the overall structure of a polynomial. Remember, being able to factor polynomials is a crucial skill that opens the door to solving more complex algebraic problems. With these factors, we could easily find the zeros of the function, which are $x = 5$ , $x = rac{1}{3}$ , and $x = -8$ . Moreover, having the polynomial in its factored form can help us in sketching the graph of the function, where the linear factors tell us the x-intercepts. Keep practicing these techniques, and you'll become a factoring pro in no time! Feel free to explore similar problems or ask questions; that's how you learn and grow. Good luck, and happy factoring!