Factoring Polynomials: A Step-by-Step Guide

by SLV Team 44 views
Factoring Polynomials: A Step-by-Step Guide

Hey guys! Let's dive into factoring the polynomial $3 s^6-1029 s^3$ completely. Factoring polynomials can seem daunting, but with a systematic approach, it becomes much easier. We'll break this down step by step, making sure everyone can follow along. So, grab your pencils, and let's get started!

1. Identifying Common Factors

First, always look for common factors in all terms of the polynomial. In our case, we have $3 s^6$ and $-1029 s^3$. We need to find the greatest common factor (GCF) of the coefficients and the variables. The GCF of 3 and 1029 is 3. For the variables, we have $s^6$ and $s^3$, so the GCF is $s^3$. Thus, we can factor out $3s^3$ from the entire polynomial.

Factoring out $3s^3$ from $3 s^6-1029 s^3$ gives us:

3s3(s3−343)3s^3(s^3 - 343)

Now, our expression looks simpler, and we can proceed with further factorization if possible. Factoring out common factors is a crucial first step because it simplifies the remaining expression, making subsequent steps easier to manage. Remember, always start by identifying and factoring out the greatest common factor.

2. Recognizing Special Patterns: Difference of Cubes

Next, let's examine the expression inside the parenthesis: $(s^3 - 343)$. Notice that both terms are perfect cubes. Specifically, $s^3$ is the cube of $s$, and $343$ is the cube of $7$ (since $7^3 = 7 \* 7 \* 7 = 343$). This means we have a difference of cubes, which follows a specific factoring pattern. The difference of cubes pattern is given by:

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2)

In our case, $a = s$ and $b = 7$. Applying the difference of cubes pattern, we get:

s3−343=(s−7)(s2+7s+49)s^3 - 343 = (s - 7)(s^2 + 7s + 49)

So, we've successfully factored the difference of cubes. Recognizing these special patterns is essential in polynomial factorization because they allow us to apply known formulas to simplify expressions quickly. Always be on the lookout for patterns like difference of squares, sum of cubes, and difference of cubes.

3. Completing the Factorization

Now that we've factored out the common factor and applied the difference of cubes pattern, let's put everything together. We started with:

3s6−1029s33 s^6-1029 s^3

We factored out $3s^3$:

3s3(s3−343)3s^3(s^3 - 343)

Then, we factored the difference of cubes:

3s3(s−7)(s2+7s+49)3s^3(s - 7)(s^2 + 7s + 49)

At this point, we need to check if the quadratic expression $(s^2 + 7s + 49)$ can be factored further. To do this, we can examine its discriminant. The discriminant, $&Delta$, of a quadratic equation $ax^2 + bx + c = 0$ is given by:

&Delta=b2−4ac\&Delta = b^2 - 4ac

In our case, $a = 1$, $b = 7$, and $c = 49$. So, the discriminant is:

&Delta=72−4(1)(49)=49−196=−147\&Delta = 7^2 - 4(1)(49) = 49 - 196 = -147

Since the discriminant is negative, the quadratic expression $(s^2 + 7s + 49)$ has no real roots and cannot be factored further using real numbers. Therefore, the completely factored form of the original polynomial is:

3s3(s−7)(s2+7s+49)3 s^3(s - 7)(s^2 + 7s + 49)

So the answer is A. $3 s3(s-7)\(s2+7 s+49)$

4. Verifying the Solution

To ensure that our factorization is correct, we can expand the factored form and see if it matches the original polynomial. Expanding $3 s^3(s - 7)(s^2 + 7s + 49)$, we get:

3s3[(s−7)(s2+7s+49)]3s^3 [(s - 7)(s^2 + 7s + 49)]

First, multiply $(s - 7)$ by $(s^2 + 7s + 49)$:

s(s2+7s+49)−7(s2+7s+49)=s3+7s2+49s−7s2−49s−343=s3−343s(s^2 + 7s + 49) - 7(s^2 + 7s + 49) = s^3 + 7s^2 + 49s - 7s^2 - 49s - 343 = s^3 - 343

Now, multiply the result by $3s^3$:

3s3(s3−343)=3s6−1029s33s^3(s^3 - 343) = 3s^6 - 1029s^3

This matches our original polynomial, so our factorization is correct. Verifying the solution by expanding the factored form is a good practice to avoid errors and build confidence in your factoring skills.

5. Common Mistakes to Avoid

When factoring polynomials, there are several common mistakes that students often make. Here are a few to watch out for:

  • Forgetting to factor out the greatest common factor (GCF): Always start by looking for the GCF in all terms of the polynomial. Failing to do so can make the subsequent steps more complicated. For example, not factoring out $3s^3$ at the beginning would leave you with larger coefficients and higher powers of $s$, making the problem harder to manage.
  • Incorrectly applying the difference of cubes pattern: Make sure you correctly identify $a$ and $b$ in the formula $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. A common mistake is to mix up the signs or the terms in the quadratic factor.
  • Assuming a quadratic expression can always be factored: Not all quadratic expressions can be factored using real numbers. Always check the discriminant to determine if a quadratic expression has real roots before attempting to factor it further.
  • Making arithmetic errors: Simple arithmetic errors can lead to incorrect factorizations. Double-check your calculations, especially when dealing with larger numbers or negative signs.
  • Not verifying the solution: Always verify your solution by expanding the factored form to ensure it matches the original polynomial. This can help you catch mistakes and build confidence in your factoring skills.

6. Advanced Factoring Techniques

For more complex polynomials, you might need to use advanced factoring techniques, such as factoring by grouping, using synthetic division, or applying the rational root theorem. Factoring by grouping involves rearranging terms and factoring out common factors from pairs of terms. Synthetic division is a shortcut method for dividing a polynomial by a linear factor. The rational root theorem helps you find potential rational roots of a polynomial, which can then be used to factor the polynomial further.

These techniques are especially useful when dealing with polynomials of higher degrees or polynomials that do not fit into the standard patterns like difference of squares or cubes. Mastering these techniques requires practice and a good understanding of polynomial properties.

7. Practice Problems

To improve your factoring skills, practice is essential. Here are a few practice problems you can try:

  1. Factor completely: $2x^3 + 16$
  2. Factor completely: $4y^6 - 32y^3$
  3. Factor completely: $5z^6 - 405z^3$

Work through these problems step by step, applying the techniques we discussed earlier. Remember to always start by factoring out the GCF, look for special patterns, and verify your solutions. The more you practice, the more comfortable and confident you will become with factoring polynomials.

Conclusion

Factoring the polynomial $3 s^6-1029 s^3$ completely involves identifying common factors, recognizing special patterns like the difference of cubes, and verifying the solution. By following a systematic approach and avoiding common mistakes, you can master polynomial factorization. Keep practicing, and you'll become a pro in no time! Happy factoring, guys!