Factoring: $-v^3 + 4v^2 + 60v$ Trinomial Made Easy

by Alex Johnson 51 views

Factoring trinomials can seem daunting, but with a systematic approach, even complex expressions like −v3+4v2+60v-v^3 + 4v^2 + 60v can be broken down into simpler components. This comprehensive guide will walk you through each step, ensuring you understand the underlying principles and can confidently tackle similar problems in the future. We'll focus on making the process clear and straightforward, so you can master factoring and apply it to various mathematical contexts. Let's dive in and unravel this trinomial together!

1. Identifying and Factoring Out the Greatest Common Factor (GCF)

When you're faced with a trinomial like −v3+4v2+60v-v^3 + 4v^2 + 60v, the first and most crucial step is to identify the Greatest Common Factor (GCF). This is the largest factor that divides each term in the trinomial. Finding the GCF simplifies the expression, making subsequent factoring steps much easier. In our example, we need to look at both the coefficients and the variable terms.

  • Coefficients: The coefficients are -1, 4, and 60. The greatest common factor for these numbers is 1. However, since the leading coefficient is negative, it’s generally good practice to factor out a -1 to make the remaining trinomial easier to work with. This means we'll factor out -1.
  • Variable Terms: The variable terms are v3v^3, v2v^2, and vv. The greatest common factor here is vv, as it is the lowest power of vv present in all terms.

Combining these, the GCF for the entire trinomial is −v-v. Factoring out −v-v from each term involves dividing each term by −v-v and writing the result in parentheses. Let’s see how this looks:

−v3+4v2+60v=−v(v2−4v−60)-v^3 + 4v^2 + 60v = -v(v^2 - 4v - 60)

Notice how factoring out −v-v changed the signs of the terms inside the parentheses. This is because we're essentially dividing each term by −v-v. Now, we have a simpler trinomial inside the parentheses, v2−4v−60v^2 - 4v - 60, which we can focus on factoring further. This initial step of factoring out the GCF is critical because it reduces the complexity of the problem and sets us up for successful factoring in the next stages. This foundational step is often overlooked but is a cornerstone of efficient and accurate trinomial factoring.

2. Factoring the Remaining Trinomial: v2−4v−60v^2 - 4v - 60

After factoring out the GCF, we're left with a simpler trinomial: v2−4v−60v^2 - 4v - 60. Now, the task is to factor this trinomial into two binomials. To do this, we need to find two numbers that satisfy two conditions:

  • Multiply to the constant term (-60 in this case).
  • Add up to the coefficient of the middle term (-4 in this case).

This process involves some strategic thinking and trial-and-error, but it becomes easier with practice. Let's break down how to approach this:

  1. List Factor Pairs of -60: Start by listing pairs of numbers that multiply to -60. Since the product is negative, one number must be positive, and the other must be negative. Here are some possibilities:

    • 1 and -60
    • -1 and 60
    • 2 and -30
    • -2 and 30
    • 3 and -20
    • -3 and 20
    • 4 and -15
    • -4 and 15
    • 5 and -12
    • -5 and 12
    • 6 and -10
    • -6 and 10

  2. Identify the Pair That Adds to -4: Now, examine each pair to see which one adds up to -4. After reviewing the list, we find that the pair 6 and -10 meet our criteria because:

    • 6 * -10 = -60
    • 6 + (-10) = -4

  3. Write the Factored Form: Once we have these numbers, we can write the trinomial in its factored form. The numbers 6 and -10 will be the constants in our two binomials:

    v2−4v−60=(v+6)(v−10)v^2 - 4v - 60 = (v + 6)(v - 10)

To double-check our work, we can use the FOIL method (First, Outer, Inner, Last) to expand the binomials and see if we get back the original trinomial:

(v+6)(v−10)=v2−10v+6v−60=v2−4v−60(v + 6)(v - 10) = v^2 - 10v + 6v - 60 = v^2 - 4v - 60

Our factored form is correct! This step of finding the right pair of numbers is crucial in factoring trinomials, and with a bit of practice, you'll become adept at identifying the correct pairs quickly. This method works efficiently for quadratic trinomials where the leading coefficient is 1. The ability to factor trinomials opens doors to solving quadratic equations and simplifying more complex algebraic expressions, making it a valuable skill in mathematics.

3. Combining the GCF with the Factored Binomials

Now that we've factored the trinomial v2−4v−60v^2 - 4v - 60 into (v+6)(v−10)(v + 6)(v - 10), we need to remember the Greatest Common Factor (GCF) we factored out in the first step. The GCF was −v-v, and it's crucial to include this in our final factored form. This step ensures we account for all factors and arrive at the completely factored expression.

To combine the GCF with the factored binomials, simply write the GCF in front of the binomial factors. This gives us the complete factored form of the original trinomial:

−v3+4v2+60v=−v(v2−4v−60)=−v(v+6)(v−10)-v^3 + 4v^2 + 60v = -v(v^2 - 4v - 60) = -v(v + 6)(v - 10)

This is the fully factored form of the given trinomial. Including the GCF is a critical step that is sometimes overlooked, but it's essential for a correct and complete answer. By factoring out the GCF first, we simplified the trinomial and made it easier to factor the remaining quadratic expression. Now, we must remember to include that GCF in the final answer. This complete factored form allows us to analyze the roots or zeros of the polynomial, which are the values of vv that make the expression equal to zero. In this case, the roots are v=0v = 0, v=−6v = -6, and v=10v = 10. Understanding how to completely factor a trinomial is not just about getting the right answer; it's also about understanding the structure of the polynomial and its behavior.

4. Verifying the Solution

After factoring a trinomial, it's always a good idea to verify your solution. This step helps ensure that you haven’t made any errors in the factoring process. There are a couple of ways you can verify your solution:

Method 1: Expanding the Factored Form

The most common method is to expand the factored form and see if it matches the original trinomial. We'll start with our factored form: −v(v+6)(v−10)-v(v + 6)(v - 10).

First, expand the two binomials (v+6)(v−10)(v + 6)(v - 10) using the FOIL method (First, Outer, Inner, Last):

(v+6)(v−10)=v2−10v+6v−60=v2−4v−60(v + 6)(v - 10) = v^2 - 10v + 6v - 60 = v^2 - 4v - 60

Now, multiply the result by the GCF, which is −v-v:

−v(v2−4v−60)=−v3+4v2+60v-v(v^2 - 4v - 60) = -v^3 + 4v^2 + 60v

This matches our original trinomial, −v3+4v2+60v-v^3 + 4v^2 + 60v, so our factoring is correct!

Method 2: Substituting Values for the Variable

Another method to verify your solution is to substitute a value for the variable (in this case, vv) into both the original trinomial and the factored form. If both expressions yield the same result, your factoring is likely correct. Let’s try substituting v=1v = 1:

  • Original Trinomial: −v3+4v2+60v-v^3 + 4v^2 + 60v

    Substitute v=1v = 1:

    −(1)3+4(1)2+60(1)=−1+4+60=63-(1)^3 + 4(1)^2 + 60(1) = -1 + 4 + 60 = 63

  • Factored Form: −v(v+6)(v−10)-v(v + 6)(v - 10)

    Substitute v=1v = 1:

    −1(1+6)(1−10)=−1(7)(−9)=63-1(1 + 6)(1 - 10) = -1(7)(-9) = 63

Both expressions evaluate to 63 when v=1v = 1, which provides further confidence in our factored form. It's good practice to try a couple of different values to increase your certainty. Verifying your solution is a crucial step in the factoring process. It not only confirms the correctness of your answer but also helps reinforce your understanding of factoring techniques. By taking the time to check your work, you can avoid common errors and build confidence in your factoring skills.

Conclusion

Factoring the trinomial −v3+4v2+60v-v^3 + 4v^2 + 60v completely involves a few key steps: identifying and factoring out the Greatest Common Factor (GCF), factoring the remaining trinomial into binomials, and then combining the GCF with the factored binomials. Finally, verifying the solution ensures accuracy. By systematically following these steps, you can confidently factor various trinomials and strengthen your algebraic skills. Remember, practice is key to mastering factoring, so keep applying these techniques to different problems.

For further learning and practice on factoring techniques, you might find helpful resources at Khan Academy Algebra. This external resource provides comprehensive lessons and practice exercises to enhance your understanding of factoring and other algebraic concepts.