Step By Step Guide To Factoring 12x^4 + 6x^3 + 18x^2

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Factoring polynomials is a fundamental skill in algebra, essential for simplifying expressions, solving equations, and understanding the behavior of functions. When we talk about completely factoring a polynomial, we mean breaking it down into its simplest factors, ideally until we can't factor it any further. This often involves identifying common factors, applying factoring patterns, and sometimes using more advanced techniques. In this comprehensive guide, we'll delve into the process of completely factoring the polynomial 12x^4 + 6x^3 + 18x^2, providing a step-by-step explanation to ensure clarity and understanding.

Identifying the Greatest Common Factor (GCF)

In factoring polynomials completely, the first and most crucial step is to identify the Greatest Common Factor (GCF). The GCF is the largest factor that divides evenly into all terms of the polynomial. To find the GCF, we need to consider both the coefficients (the numerical parts) and the variables (the literal parts) of each term. For the polynomial 12x^4 + 6x^3 + 18x^2, let's break down the process:

  1. Coefficients: We have the coefficients 12, 6, and 18. The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 6 are 1, 2, 3, and 6. The factors of 18 are 1, 2, 3, 6, 9, and 18. The greatest common factor among these numbers is 6.
  2. Variables: We have the variable terms x^4, x^3, and x^2. The GCF for variables is the lowest power of the common variable present in all terms. In this case, the lowest power of x is x^2.

Combining these findings, the GCF of the polynomial 12x^4 + 6x^3 + 18x^2 is 6x^2. This means that 6x^2 can be factored out from each term in the polynomial. Factoring out the GCF is a crucial step because it simplifies the polynomial and makes subsequent factoring steps easier. After identifying the GCF, we proceed to the next step, which involves dividing each term of the original polynomial by the GCF.

Factoring out the GCF

Once we have identified the Greatest Common Factor (GCF), the next step in factoring polynomials completely is to factor it out from the original polynomial. This involves dividing each term of the polynomial by the GCF and writing the polynomial as a product of the GCF and the resulting expression. In our case, the polynomial is 12x^4 + 6x^3 + 18x^2, and the GCF we found is 6x^2. To factor out the GCF, we perform the following steps:

  1. Divide each term by the GCF:
    • 12x^4 / (6x^2) = 2x^(4-2) = 2x^2
    • 6x^3 / (6x^2) = x^(3-2) = x
    • 18x^2 / (6x^2) = 3
  2. Write the factored expression: Now, we write the original polynomial as the product of the GCF and the expression obtained by dividing each term by the GCF. This gives us:
    • 12x^4 + 6x^3 + 18x^2 = 6x2(2x2 + x + 3)

What we have done here is to rewrite the polynomial in a factored form. The expression 6x2(2x2 + x + 3) is equivalent to the original polynomial, but it is now expressed as a product. This is a significant step in factoring polynomials completely because it simplifies the polynomial and makes it easier to analyze and solve. The next step is to examine the remaining expression within the parentheses (2x^2 + x + 3) to see if it can be factored further. This involves looking for factoring patterns and applying appropriate techniques.

Examining the Remaining Quadratic Expression

After factoring out the Greatest Common Factor (GCF), the next crucial step in factoring polynomials completely is to examine the remaining expression to see if it can be factored further. In our example, after factoring out 6x^2 from 12x^4 + 6x^3 + 18x^2, we were left with the quadratic expression 2x^2 + x + 3. This is a quadratic expression because it is a polynomial of degree two. Factoring a quadratic expression involves finding two binomials (expressions with two terms) that, when multiplied together, give the quadratic expression.

To determine if 2x^2 + x + 3 can be factored further, we can use several methods. One common method is to look for two numbers that multiply to give the product of the leading coefficient (2) and the constant term (3), which is 2 * 3 = 6, and add up to the middle coefficient (1). In other words, we are looking for two numbers that multiply to 6 and add to 1. However, after considering the factor pairs of 6 (1 and 6, 2 and 3), it becomes clear that there are no such integer numbers that satisfy these conditions. Another method to check if a quadratic expression can be factored is to compute the discriminant. The discriminant, denoted as Δ, is given by the formula Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic expression ax^2 + bx + c. For the expression 2x^2 + x + 3, a = 2, b = 1, and c = 3. The discriminant is:

Δ = 1^2 - 4 * 2 * 3 = 1 - 24 = -23

Since the discriminant is negative, this indicates that the quadratic expression 2x^2 + x + 3 has no real roots and cannot be factored further using real numbers. This is a critical determination in factoring polynomials completely because it tells us that we have reached the simplest form of the polynomial. If the quadratic expression could be factored, we would proceed to write it as a product of two binomials. However, in this case, since it cannot be factored further, we can conclude that the factored form we obtained in the previous step is the complete factorization.

Final Factored Form

Having identified and factored out the Greatest Common Factor (GCF), and after thoroughly examining the remaining quadratic expression, we can now confidently state the final factored form of the original polynomial. In factoring polynomials completely, the goal is to break down the polynomial into its simplest factors. For the polynomial 12x^4 + 6x^3 + 18x^2, we first identified the GCF as 6x^2 and factored it out, resulting in the expression:

6x2(2x2 + x + 3)

Next, we examined the quadratic expression 2x^2 + x + 3 to determine if it could be factored further. By attempting to find two numbers that multiply to 6 (the product of the leading coefficient and the constant term) and add to 1 (the middle coefficient), we found that no such integer numbers exist. Furthermore, by computing the discriminant (Δ = b^2 - 4ac), we found that Δ = -23, which is negative. This confirms that the quadratic expression has no real roots and cannot be factored further using real numbers. Therefore, the expression 2x^2 + x + 3 is prime, meaning it cannot be factored into simpler polynomials with real coefficients. Consequently, the completely factored form of the polynomial 12x^4 + 6x^3 + 18x^2 is:

6x2(2x2 + x + 3)

This is the final answer, as we have successfully broken down the original polynomial into its simplest factors. The expression 6x^2 is a monomial factor, and (2x^2 + x + 3) is a prime quadratic factor. This thorough process demonstrates factoring polynomials completely, ensuring that no further factoring is possible. By understanding and applying these steps, one can efficiently and accurately factor various polynomials.

Conclusion

In conclusion, factoring polynomials completely is a vital skill in algebra that involves breaking down a polynomial into its simplest factors. For the polynomial 12x^4 + 6x^3 + 18x^2, we systematically applied the steps of factoring, which included identifying and factoring out the Greatest Common Factor (GCF), and then examining the remaining quadratic expression for further factorization. We successfully identified the GCF as 6x^2 and factored it out, resulting in 6x2(2x2 + x + 3). Upon examining the quadratic expression 2x^2 + x + 3, we determined that it could not be factored further using real numbers, as its discriminant was negative. This led us to the final completely factored form:

6x2(2x2 + x + 3)

This process highlights the importance of a step-by-step approach in factoring polynomials completely. By first identifying and factoring out the GCF, we simplify the polynomial, making subsequent steps easier. Then, by carefully examining the remaining expression and applying appropriate techniques, such as checking the discriminant for quadratic expressions, we ensure that the polynomial is factored to its simplest form. Mastering these techniques not only aids in solving algebraic problems but also enhances one’s understanding of polynomial structures and their properties. Factoring is a fundamental tool in algebra, with applications in various areas of mathematics and beyond. By practicing and understanding these methods, one can confidently tackle a wide range of factoring problems.