Greatest Common Factor Of Polynomials Factoring \(44k^5 - 66k^4 + 77k^3\)

Hey everyone! Today, we're diving into the fascinating world of polynomials and tackling a common challenge: finding the greatest common factor (GCF) and factoring polynomials completely. We'll be working with the expression ${44k^5 - 66k^4 + 77k^3}$, and by the end of this article, you'll be a pro at cracking similar problems. Let's get started!

Understanding the Greatest Common Factor (GCF)

Before we jump into the specifics of our polynomial, let's make sure we're all on the same page about what the greatest common factor actually means. In simple terms, the GCF is the largest factor that divides evenly into two or more numbers or terms. Think of it as the biggest piece you can pull out of a puzzle that fits into all the relevant spots.

When dealing with polynomials, the GCF can involve both numerical coefficients and variable terms. To find the GCF, we break down each term into its prime factors and identify the factors they share. The product of these shared factors, raised to the lowest power they appear in any of the terms, gives us the GCF. This process might sound a bit intimidating at first, but trust me, it becomes second nature with practice. Remember, understanding the greatest common factor is like having a superpower when it comes to simplifying and solving algebraic expressions.

For example, if we have the numbers 12 and 18, their prime factorizations are:

• 12 = 2 x 2 x 3
• 18 = 2 x 3 x 3

The common factors are 2 and 3. Multiplying these together, we get 2 x 3 = 6, which is the GCF of 12 and 18. We'll apply a similar approach to find the GCF of the terms in our polynomial.

Finding the GCF of ${44k^5 - 66k^4 + 77k^3}$

Okay, now let's get our hands dirty with the polynomial ${44k^5 - 66k^4 + 77k^3}$. Our first mission is to identify the GCF of the three terms: ${44k^5}$, ${-66k^4}$, and ${77k^3}$. To do this, we'll break down the coefficients and variable parts separately.

Breaking Down the Coefficients

Let's start with the coefficients: 44, -66, and 77. We need to find the largest number that divides evenly into all three. This might require a bit of mental math or a quick prime factorization, but don't worry, we've got this!

Think about the factors of each number:

• Factors of 44: 1, 2, 4, 11, 22, 44
• Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66
• Factors of 77: 1, 7, 11, 77

Looking at these factors, we can see that the greatest common factor of 44, 66, and 77 is 11. So, we've got the numerical part of our GCF – it's 11!

Analyzing the Variable Terms

Now, let's tackle the variable parts: ${k^5}$, ${k^4}$, and ${k^3}$. Remember, when finding the GCF of variable terms, we take the variable with the smallest exponent. Why? Because that's the highest power of the variable that is common to all the terms.

In this case, we have ${k^5}$ (which is k multiplied by itself five times), ${k^4}$ (k multiplied by itself four times), and ${k^3}$ (k multiplied by itself three times). The smallest exponent is 3, so the variable part of our GCF is ${k^3}$. This means that each term in the original polynomial is divisible by ${k^3}$.

Putting It All Together

We've found that the greatest common factor of the coefficients is 11, and the GCF of the variable terms is ${k^3}$. To get the complete GCF of the polynomial, we simply multiply these together. So, the GCF of ${44k^5 - 66k^4 + 77k^3}$ is ${11k^3}$. Awesome! We've conquered the first major step.

Factoring Out the GCF

Now that we've identified the GCF, the next step is to factor it out of the polynomial. Factoring is like reverse distribution – we're essentially pulling out the common factor and seeing what's left behind. This is where the magic happens, guys!

To factor out the GCF, we divide each term in the polynomial by the GCF we found. Remember, we're dividing both the coefficients and the variable terms.

Here's how it works:

1. Divide ${44k^5}$ by ${11k^3}$:
   • 44 ÷ 11 = 4
   • ${k^5}$ ÷ ${k^3}$ = ${k^(5-3)}$ = ${k^2}$
   • So, ${44k^5}$ ÷ ${11k^3}$ = ${4k^2}$
2. Divide ${-66k^4}$ by ${11k^3}$:
   • -66 ÷ 11 = -6
   • ${k^4}$ ÷ ${k^3}$ = ${k^(4-3)}$ = ${k}$
   • So, ${-66k^4}$ ÷ ${11k^3}$ = ${-6k}$
3. Divide ${77k^3}$ by ${11k^3}$:
   • 77 ÷ 11 = 7
   • ${k^3}$ ÷ ${k^3}$ = 1 (since anything divided by itself is 1)
   • So, ${77k^3}$ ÷ ${11k^3}$ = 7

Now, we take the results of these divisions and put them inside parentheses. The GCF goes outside the parentheses. This gives us the factored form of the polynomial:

${11k^3(4k^2 - 6k + 7)}$

Verifying the Factored Form

To make sure we've factored correctly, it's always a good idea to check our work. We can do this by distributing the GCF back into the parentheses. If we end up with the original polynomial, we know we're on the right track.

Let's distribute ${11k^3}$ into ${(4k^2 - 6k + 7)}$:

• ${11k^3 * 4k^2 = 44k^5}$
• ${11k^3 * -6k = -66k^4}$
• ${11k^3 * 7 = 77k^3}$

Adding these terms together, we get ${44k^5 - 66k^4 + 77k^3}$, which is exactly our original polynomial. High five! We've successfully factored out the GCF and verified our answer.

The Complete Factorization

So, what's the final verdict? We've found that the greatest common factor of the polynomial ${44k^5 - 66k^4 + 77k^3}$ is ${11k^3}$, and the completely factored form is:

${11k^3(4k^2 - 6k + 7)}$

This corresponds to option A) in the original question. You nailed it!

Diving Deeper: Can We Factor Further?

Now, a crucial question to ask ourselves is: can we factor the expression inside the parentheses, ${(4k^2 - 6k + 7)}$, any further? This is a quadratic expression, and there are a few ways we could try to factor it. We could look for two numbers that multiply to (4 * 7 = 28) and add up to -6, but after a bit of thought, you'll realize that no such integer numbers exist. Another approach would be to use the quadratic formula to find the roots of the equation ${4k^2 - 6k + 7 = 0}$, but you'll find that the discriminant (the part under the square root) is negative, meaning the roots are complex numbers.

For the purpose of this exercise and within the realm of factoring with integer coefficients, we can conclude that ${(4k^2 - 6k + 7)}$ is irreducible, meaning it cannot be factored further using integers. Therefore, the complete factorization of the original polynomial is indeed ${11k^3(4k^2 - 6k + 7)}$.

Why Factoring Matters

You might be wondering, "Why bother with factoring at all?" Well, guys, factoring is a fundamental skill in algebra and has tons of applications. It's like having a Swiss Army knife for solving mathematical problems.

Factoring helps us:

• Simplify expressions: By factoring, we can rewrite complex expressions in a simpler form, making them easier to work with.
• Solve equations: Factoring is a key technique for solving polynomial equations. When we factor an equation, we can set each factor equal to zero and find the solutions.
• Analyze graphs: Factoring can help us find the x-intercepts (also called roots or zeros) of a polynomial function, which are important points on the graph.
• Work with rational expressions: Factoring is essential for simplifying, adding, subtracting, multiplying, and dividing rational expressions (fractions with polynomials in the numerator and denominator).

So, mastering factoring is definitely worth the effort. It's a skill that will serve you well throughout your math journey.

Practice Makes Perfect

The best way to become a factoring whiz is to practice, practice, practice! Try working through more examples, and don't be afraid to make mistakes – that's how we learn. Remember, practice is the most perfect way to master factoring. The more you practice, the more comfortable and confident you'll become.

Conclusion: You're a Factoring Pro!

Alright, guys! We've covered a lot of ground today. We've learned how to find the greatest common factor of a polynomial, how to factor it out, and how to verify our answer. We've also discussed why factoring is so important in mathematics. You've now got the tools you need to tackle similar problems with confidence. Keep up the great work, and happy factoring!

Remember, the key to mastering any math concept is to understand the fundamentals, practice regularly, and don't be afraid to ask for help when you need it. You've got this!