Complete Factorization Of X^4 - 8x^2 + 16 A Step-by-Step Guide

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Factoring polynomials is a fundamental skill in algebra, and mastering it opens doors to solving complex equations and understanding the behavior of functions. In this comprehensive guide, we will delve into the complete factorization of the polynomial x4 - 8_x_2 + 16, ensuring that all factors have integer coefficients. This problem showcases a classic example of how recognizing patterns and applying algebraic identities can simplify seemingly complex expressions. We will break down the steps involved, providing clear explanations and insights along the way. So, whether you're a student looking to solidify your understanding or someone seeking a refresher on factoring techniques, this article will equip you with the knowledge and confidence to tackle similar problems.

Recognizing the Pattern: A Perfect Square Trinomial

The initial key to factoring x4 - 8_x_2 + 16 lies in recognizing its structure. At first glance, it might seem intimidating due to the presence of the x4 term. However, a closer look reveals a pattern that aligns with a perfect square trinomial. A perfect square trinomial is a trinomial that can be expressed as the square of a binomial. The general form of a perfect square trinomial is a2 ± 2*ab* + b2, which can be factored as (a ± b)2. To effectively identify a perfect square trinomial, focus on a few key features. First, the first and last terms should be perfect squares themselves. This means that they can be written as the square of some expression. In our case, x4 is the square of x2, and 16 is the square of 4. Second, the middle term should be twice the product of the square roots of the first and last terms. Here, the square root of x4 is x2, and the square root of 16 is 4. Twice their product is 2 * x2 * 4 = 8_x_2, which matches the absolute value of our middle term. This confirms that our polynomial, x4 - 8_x_2 + 16, indeed fits the pattern of a perfect square trinomial. The negative sign in front of the 8_x_2 term indicates that we will use the subtraction form of the perfect square trinomial identity, which is (a - b)2 = a2 - 2*ab* + b2. By recognizing this pattern, we can significantly simplify the factorization process. Instead of resorting to more complex methods, we can directly apply the perfect square trinomial identity to break down the polynomial into a more manageable form. This pattern recognition is a crucial skill in algebra, allowing for efficient problem-solving and a deeper understanding of polynomial structures. It transforms what might initially appear as a daunting task into a straightforward application of a known identity.

Applying the Perfect Square Trinomial Identity

Once we've identified x4 - 8_x_2 + 16 as a perfect square trinomial, the next step is to apply the corresponding identity to begin the factorization process. The perfect square trinomial identity, as mentioned earlier, is given by a2 - 2*ab* + b2 = (a - b)2. This identity provides a direct pathway to factoring expressions that fit this specific pattern. In our polynomial, we can map the terms to the identity as follows: x4 corresponds to a2, -8_x_2 corresponds to -2*ab, and 16 corresponds to b2. This mapping allows us to determine the values of a and b. Since x4 = a2, we can deduce that a = x2. Similarly, since 16 = b2, we find that b = 4. Now that we have the values of a and b, we can substitute them into the factored form of the perfect square trinomial identity, which is (a* - b)2. Substituting a = x2 and b = 4, we get (x2 - 4)2. This substitution represents a significant step in the factorization process. We've successfully transformed the original fourth-degree polynomial into a squared expression involving a quadratic term. However, our factorization journey is not yet complete. While we've simplified the expression, we must ensure that all factors have integer coefficients and that the factorization is complete. The expression (x2 - 4)2 indicates that (x2 - 4) is a factor, but we need to examine it further to see if it can be factored further. This leads us to the next crucial step in the factorization process: recognizing and applying the difference of squares pattern.

Recognizing and Applying the Difference of Squares

After applying the perfect square trinomial identity, we arrived at the expression (x2 - 4)2. To completely factor x4 - 8_x_2 + 16, we must now focus on the term inside the parentheses, x2 - 4. This expression is not a prime factor; it can be further factored by recognizing another fundamental algebraic pattern: the difference of squares. The difference of squares is a pattern where a binomial is formed by subtracting one perfect square from another. The general form of the difference of squares is a2 - b2, which can be factored as (a + b) (a - b). This identity is a powerful tool for factoring expressions that fit this pattern, and it's essential for completely factoring polynomials. In our case, x2 - 4 perfectly matches the difference of squares pattern. We can identify x2 as a2 and 4 as b2. This allows us to determine the values of a and b. Since x2 = a2, we have a = x. Similarly, since 4 = b2, we have b = 2. Now that we have identified a and b, we can apply the difference of squares identity to factor x2 - 4. Substituting a = x and b = 2 into the factored form (a + b) (a - b), we get (x + 2) (x - 2). This factorization is a critical step in completely factoring the original polynomial. We've broken down x2 - 4 into two linear factors, (x + 2) and (x - 2). These factors are linear because the highest power of x in each factor is 1. Now, we need to incorporate this factorization back into our previous expression, (x2 - 4)2, to obtain the complete factorization of the original polynomial. This involves replacing (x2 - 4) with its factored form, (x + 2) (x - 2), and then considering the square that was applied to the entire expression. This final step will give us the fully factored form of x4 - 8_x_2 + 16, with all factors having integer coefficients.

Completing the Factorization

Having successfully factored x2 - 4 into (x + 2)(x - 2), we can now complete the factorization of x4 - 8_x_2 + 16. Recall that we previously simplified the polynomial to (x2 - 4)2 using the perfect square trinomial identity. This means that the expression (x2 - 4) is squared, indicating that it is multiplied by itself. To incorporate the difference of squares factorization, we substitute (x + 2)(x - 2) for (x2 - 4) in the expression (x2 - 4)2. This gives us [(x + 2)(x - 2)]2. Now, we can apply the exponent to each factor within the brackets. When a product is raised to a power, each factor in the product is raised to that power. Therefore, [(x + 2)(x - 2)]2 becomes (x + 2)2(x - 2)2. This expression represents the complete factorization of x4 - 8_x_2 + 16. We have successfully broken down the original fourth-degree polynomial into a product of linear factors, each raised to a power. The factors are (x + 2) and (x - 2), and they are both squared. This indicates that each factor appears twice in the complete factorization. The final factored form, (x + 2)2(x - 2)2, satisfies the requirement that all factors have integer coefficients. The coefficients in the factors (x + 2) and (x - 2) are all integers (1, 2, -2). Moreover, we have factored the polynomial completely, as there are no further factorizations possible with integer coefficients. Each factor is a linear term, and linear terms are irreducible over the integers. This complete factorization provides valuable information about the roots of the polynomial equation x4 - 8_x_2 + 16 = 0. The roots are the values of x that make the polynomial equal to zero. From the factored form, we can see that the roots are x = -2 and x = 2. Each of these roots has a multiplicity of 2, which means that they appear twice as solutions to the equation. This is because the factors (x + 2) and (x - 2) are squared in the complete factorization.

Final Answer

In conclusion, the complete factorization of the polynomial x4 - 8_x_2 + 16, with all factors having integer coefficients, is (x + 2)2(x - 2)2. This factorization was achieved by recognizing the polynomial as a perfect square trinomial and then applying the difference of squares identity. The process involved several key steps: identifying the perfect square trinomial pattern, applying the corresponding identity to simplify the expression, recognizing the difference of squares pattern within the simplified expression, applying the difference of squares identity to factor further, and finally, combining the results to obtain the complete factorization. This problem exemplifies the importance of recognizing algebraic patterns and applying appropriate identities to simplify and factor polynomials. The ability to identify patterns like perfect square trinomials and the difference of squares is a fundamental skill in algebra, enabling efficient problem-solving and a deeper understanding of mathematical structures. The factored form, (x + 2)2(x - 2)2, not only provides a concise representation of the polynomial but also reveals important information about its roots and behavior. The roots of the polynomial equation x4 - 8_x_2 + 16 = 0 are x = -2 and x = 2, each with a multiplicity of 2. This means that the graph of the polynomial function y = x4 - 8_x_2 + 16 touches the x-axis at x = -2 and x = 2 but does not cross it. The complete factorization also allows us to analyze the polynomial's degree, leading coefficient, and end behavior. The degree of the polynomial is 4, which is the highest power of x in the expression. The leading coefficient is 1, which is the coefficient of the x4 term. Since the degree is even and the leading coefficient is positive, the graph of the polynomial function rises to infinity as x approaches both positive and negative infinity. This comprehensive factorization process and the analysis of the factored form demonstrate the power and utility of algebraic techniques in understanding and manipulating polynomial expressions.