Factored Polynomials: Which Expression Is Fully Factored?

by Rajiv Sharma 58 views

Hey guys! Today, we're diving into the fascinating world of polynomials and factorization. Factoring polynomials is a fundamental skill in algebra, and it's super important for simplifying expressions, solving equations, and understanding more advanced math concepts. We're going to break down what it means for a polynomial to be factored completely and then walk through a specific example to help you master this skill. So, let's get started!

What Does "Factored Completely" Really Mean?

Before we jump into our example, let's make sure we're all on the same page about what it means for a polynomial to be factored completely. Imagine you're taking apart a Lego structure. Factoring is like taking a polynomial apart into its building blocks, which are smaller polynomials or monomials. To be factored completely, it means you've taken it apart as much as possible—you can't break it down any further.

Think of it this way: you're looking for the smallest possible "pieces" that multiply together to give you the original polynomial. These pieces are called factors. A polynomial is factored completely when each factor cannot be factored any further. This often means you've expressed the polynomial as a product of prime factors, much like how you can break down a number into its prime factors (e.g., 12 = 2 x 2 x 3).

Key Indicators of Complete Factorization

So, how do you know when you've factored a polynomial completely? Here are a few key things to look for:

  1. Greatest Common Factor (GCF): Have you taken out the greatest common factor from all terms? The GCF is the largest factor that divides all terms in the polynomial. If you haven't factored out the GCF, you haven't factored completely.
  2. Difference of Squares: Do you see any factors in the form of a² - b²? This can be factored into (a + b)(a - b). If you miss this, you haven't factored completely.
  3. Perfect Square Trinomials: Do you see any factors in the form of a² + 2ab + b² or a² - 2ab + b²? These can be factored into (a + b)² or (a - b)², respectively. Missing these means incomplete factorization.
  4. Other Factorable Forms: Can you factor any quadratic expressions (ax² + bx + c) further? This often involves finding two numbers that multiply to ac and add up to b. If you can still factor a quadratic, you're not done yet.

In essence, a polynomial is factored completely when you've exhausted all possible factoring techniques and cannot break down any of the factors any further. It’s like simplifying a fraction to its lowest terms—you keep going until you can’t simplify anymore.

Analyzing the Given Options: A Step-by-Step Breakdown

Now, let's apply this understanding to the options you've provided. Our goal is to identify which polynomial is factored completely. We'll examine each option closely, using the criteria we discussed earlier.

Option A: 4(4x⁴ - 1)

At first glance, this might seem factored, but let's dig a little deeper. The expression inside the parentheses, 4x⁴ - 1, looks like it might be factorable further. Specifically, it resembles a difference of squares. Remember, the difference of squares pattern is a² - b² = (a + b)(a - b). Can we rewrite 4x⁴ - 1 in this form?

Yes, we can! Notice that 4x⁴ is (2x²)² and 1 is 1². So, we can rewrite 4x⁴ - 1 as (2x²)² - 1². Applying the difference of squares pattern, we get:

4x⁴ - 1 = (2x² + 1)(2x² - 1)

Now, let's look at the factors we've obtained. The term (2x² + 1) cannot be factored further using real numbers. However, the term (2x² - 1) also looks like a difference of squares, although it might not be immediately obvious. We can rewrite it as (√2x)² - 1².

Applying the difference of squares pattern again, we get:

2x² - 1 = (√2x + 1)(√2x - 1)

Thus, the complete factorization of 4(4x⁴ - 1) is:

4(2x² + 1)(√2x + 1)(√2x - 1)

Since Option A can be factored further, it is not the polynomial that is factored completely.

Option B: 2x(y³ - 4y² + 5y)

Let’s take a look at Option B: 2x(y³ - 4y² + 5y). The first thing we always want to check is if there's a greatest common factor (GCF) within the polynomial inside the parentheses. In this case, we have y³ - 4y² + 5y. Notice that each term has a factor of y. So, we can factor out a y from the expression inside the parentheses:

y³ - 4y² + 5y = y(y² - 4y + 5)

Now, our expression looks like this:

2x * y(y² - 4y + 5)

Or, simplifying a bit:

2xy(y² - 4y + 5)

Next, we need to see if the quadratic expression (y² - 4y + 5) can be factored further. To do this, we look for two numbers that multiply to 5 (the constant term) and add up to -4 (the coefficient of the y term). The factors of 5 are 1 and 5. However, there are no integer factors of 5 that add up to -4. This means that the quadratic y² - 4y + 5 cannot be factored further using integers. We could check the discriminant (b² - 4ac) to confirm whether it has real roots, but for the purpose of complete factorization over integers, we can conclude it's not factorable further.

However, we did find a common factor y within the parenthesis. Therefore, Option B can still be factored, meaning it is not factored completely in its original form.

Option C: 3(52 + 1)

Option C presents us with 3(52 + 1). Let's simplify the expression inside the parentheses first:

52 + 1 = 25 + 1 = 26

So, the expression becomes:

3(26)

Now, let's evaluate this:

3 * 26 = 78

The expression 78 is just a number, not a polynomial. While we can find the prime factorization of 78 (which is 2 * 3 * 13), the question asks which polynomial is factored completely. Since 78 is not a polynomial, Option C is not the correct answer. This option seems to be a bit of a distraction, as it doesn't involve polynomial factorization at all.

Option D: 5x² - 17x + 14

Now, let's examine Option D: 5x² - 17x + 14. This is a quadratic expression, and we need to determine if it can be factored further. To factor a quadratic in the form ax² + bx + c, we look for two numbers that multiply to ac (in this case, 5 * 14 = 70) and add up to b (which is -17).

Let's list the factor pairs of 70:

  • 1 and 70
  • 2 and 35
  • 5 and 14
  • 7 and 10

We need a pair that adds up to -17. Notice that -7 and -10 satisfy this condition since (-7) * (-10) = 70 and (-7) + (-10) = -17. Now, we can use these numbers to factor the quadratic expression. We'll use the factoring by grouping method:

  1. Rewrite the middle term using the two numbers we found:

    5x² - 17x + 14 = 5x² - 7x - 10x + 14

  2. Group the terms:

    (5x² - 7x) + (-10x + 14)

  3. Factor out the GCF from each group:

    x(5x - 7) - 2(5x - 7)

  4. Notice that (5x - 7) is a common factor. Factor it out:

    (5x - 7)(x - 2)

So, we have factored 5x² - 17x + 14 into (5x - 7)(x - 2). Now, we need to check if these factors can be factored further. Both (5x - 7) and (x - 2) are linear expressions, and they cannot be factored further. Therefore, the polynomial 5x² - 17x + 14 is factored completely as (5x - 7)(x - 2).

The Verdict: Which Polynomial is Factored Completely?

After our detailed analysis, we've determined that:

  • Option A can be factored further using the difference of squares pattern.
  • Option B has a common factor that can be factored out.
  • Option C is not a polynomial factorization question.
  • Option D is factored into two linear expressions that cannot be factored further.

Therefore, the correct answer is Option D: 5x² - 17x + 14.

Key Takeaways and Final Thoughts

Alright, guys, we've covered a lot in this guide! We started by defining what it means for a polynomial to be factored completely—breaking it down into its simplest factors. We looked at key indicators like the greatest common factor, difference of squares, and perfect square trinomials.

Then, we applied these concepts to the given options, carefully analyzing each one. We saw how Option A could be factored further using the difference of squares, how Option B had a common factor, and how Option C wasn't even a polynomial factorization question. Finally, we successfully factored Option D and confirmed that it was indeed factored completely.

Factoring polynomials is a crucial skill in algebra, and mastering it will help you tackle more complex problems with confidence. Remember to always look for the GCF first, and then consider other factoring techniques like difference of squares, perfect square trinomials, and factoring quadratics. Keep practicing, and you'll become a factoring pro in no time!

If you have any more questions or want to explore other math topics, feel free to ask. Keep up the great work, and happy factoring!