Factoring Polynomials: How To Factor X^2 - 16x + 48

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Hey guys! Today, we're diving into the fascinating world of polynomial factorization. Specifically, we're going to break down the polynomial x^2 - 16x + 48 into its factored form. Factoring polynomials is a crucial skill in algebra, and it pops up everywhere from solving equations to simplifying expressions. So, let's get started and make sure we understand every step of the process. Grab your pencils and paper, and let’s jump right in!

Understanding Polynomial Factorization

Before we tackle our specific problem, let's quickly recap what polynomial factorization actually means. Think of it as the reverse of expanding brackets. When we expand, we multiply terms together to get a polynomial. Factoring is the process of finding the expressions that, when multiplied, give us the original polynomial. In simpler terms, we're trying to find the building blocks of our polynomial.

For a quadratic polynomial like x^2 - 16x + 48, we're looking for two binomials (expressions with two terms) that, when multiplied, give us the original quadratic. This often takes the form of (x + a)(x + b), where 'a' and 'b' are constants. Our mission is to find the right values for 'a' and 'b'.

Why is this important? Well, factored form makes it much easier to solve equations. For instance, if we have (x - 4)(x - 12) = 0, we immediately know that either x - 4 = 0 or x - 12 = 0, giving us solutions x = 4 and x = 12. This is why mastering factorization is so vital for tackling more advanced algebraic problems. Factoring isn't just a random mathematical exercise; it's a powerful tool that unlocks solutions and simplifies complex expressions, making your mathematical journey smoother and more efficient. Think of it as learning a new language – once you understand the grammar (in this case, the rules of factoring), you can communicate more effectively (solve problems more easily).

The Factoring Process: A Step-by-Step Guide

Now, let’s get to the nitty-gritty of factoring x^2 - 16x + 48. We’ll break it down into simple, manageable steps so you can follow along easily.

Step 1: Identify the Coefficients

First, we need to identify the coefficients in our quadratic polynomial. A quadratic polynomial generally looks like ax^2 + bx + c. In our case, x^2 - 16x + 48, we have:

  • a = 1 (the coefficient of x^2)
  • b = -16 (the coefficient of x)
  • c = 48 (the constant term)

These coefficients are our key players, and we'll use them to find the factors.

Step 2: Find Two Numbers

This is the heart of the factoring process. We need to find two numbers that:

  1. Multiply to give 'c' (which is 48)
  2. Add up to give 'b' (which is -16)

This might sound tricky, but there's a systematic way to approach it. Start by listing the factor pairs of 48:

  • 1 and 48
  • 2 and 24
  • 3 and 16
  • 4 and 12
  • 6 and 8

Now, remember that we need these two numbers to add up to -16. This means both numbers must be negative, since a negative plus a negative gives us a negative. Looking at our pairs, we can see that -4 and -12 fit the bill:

  • (-4) * (-12) = 48
  • (-4) + (-12) = -16

Bingo! We've found our numbers. This step is like detective work; you're piecing together clues to find the right combination. Don't be discouraged if it takes a few tries – practice makes perfect!

Step 3: Write the Factored Form

Now that we have our two numbers, -4 and -12, we can write the factored form of the polynomial. It's super simple:

x^2 - 16x + 48 = (x - 4)(x - 12)

See how the numbers we found, -4 and -12, appear in the binomials? That’s the magic of factoring! We've successfully transformed our original polynomial into a product of two simpler expressions.

Step 4: Verify (Optional but Recommended)

To make sure we’ve got it right, we can expand our factored form and see if it matches the original polynomial. Let's use the FOIL method (First, Outer, Inner, Last):

(x - 4)(x - 12) = xx + x(-12) + (-4)x + (-4)(-12)

= x^2 - 12x - 4x + 48

= x^2 - 16x + 48

It matches! This step is like the final checkmark, ensuring you’ve nailed it. Verifying your answer builds confidence and helps reinforce the process in your mind.

Applying the Steps to Our Problem: x^2 - 16x + 48

Let's quickly recap how we applied these steps to factor x^2 - 16x + 48:

  1. Identify Coefficients: a = 1, b = -16, c = 48
  2. Find Two Numbers: -4 and -12 (because -4 * -12 = 48 and -4 + -12 = -16)
  3. Write Factored Form: (x - 4)(x - 12)
  4. Verify: (x - 4)(x - 12) expands to x^2 - 16x + 48

So, the factored form of x^2 - 16x + 48 is indeed (x - 4)(x - 12).

Why This Method Works

You might be wondering, why does this method work? It all comes down to the distributive property of multiplication. When we expand (x + a)(x + b), we get:

(x + a)(x + b) = x^2 + bx + ax + ab

= x^2 + (a + b)x + ab

Notice that the coefficient of x is the sum of 'a' and 'b', and the constant term is the product of 'a' and 'b'. This is why we look for two numbers that add up to 'b' and multiply to 'c'. Understanding this connection helps you see the logic behind the process, making it easier to remember and apply.

Common Mistakes to Avoid

Factoring can sometimes be tricky, and it’s easy to make small errors. Here are a few common mistakes to watch out for:

  1. Sign Errors: Pay close attention to the signs of the numbers. A simple mistake with a negative sign can throw off the whole factorization.
  2. Incorrect Factor Pairs: Make sure you've considered all possible factor pairs before settling on a solution.
  3. Forgetting to Verify: Always verify your answer by expanding the factored form. This helps catch mistakes early.
  4. Stopping Too Early: Sometimes, you might find two numbers that multiply to 'c' but don't add up to 'b'. Keep searching!

Practice Problems

To really master factoring, practice is key! Here are a few problems for you to try:

  1. Factor x^2 - 10x + 24
  2. Factor x^2 + 5x - 14
  3. Factor x^2 - 2x - 15

Work through these problems using the steps we've discussed, and don't forget to verify your answers. The more you practice, the more confident you'll become in your factoring skills.

Conclusion

So there you have it! We've walked through the process of factoring the polynomial x^2 - 16x + 48 step by step. Remember, factoring is a fundamental skill in algebra, and with practice, you can become a pro. Don't be afraid to tackle more complex polynomials – you've got the tools and knowledge to succeed. Keep practicing, keep learning, and you'll be factoring like a champ in no time!

If you found this guide helpful, give it a thumbs up and share it with your friends. And if you have any questions or want to see more examples, drop a comment below. Happy factoring, guys!