Completely Factored Form Of 3x⁵ - 7x⁴ + 6x² - 14x

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In the realm of algebra, factoring polynomials stands as a fundamental skill, enabling us to simplify complex expressions and solve intricate equations. When confronted with the polynomial 3x⁵ - 7x⁴ + 6x² - 14x, the task at hand is to determine its completely factored form. This involves breaking down the polynomial into its simplest multiplicative components, revealing its underlying structure and making it easier to analyze. Let's delve into the process of unraveling this polynomial and expressing it in its most reduced form.

Unveiling the Initial Factors

Our journey begins with the given polynomial: 3x⁵ - 7x⁴ + 6x² - 14x. The first step in factoring any polynomial is to identify any common factors that can be extracted from all terms. In this case, we observe that 'x' is a common factor present in each term. Factoring out 'x', we get:

x(3x⁴ - 7x³ + 6x - 14)

This initial factorization simplifies the polynomial, reducing the degree of the expression within the parentheses. However, our quest for the completely factored form is far from over. We must now focus on factoring the expression within the parentheses: 3x⁴ - 7x³ + 6x - 14. This quartic polynomial (a polynomial of degree four) presents a more challenging factoring puzzle.

Delving Deeper: Factoring by Grouping

When faced with polynomials containing four or more terms, a powerful technique known as factoring by grouping often comes to the rescue. This method involves strategically grouping terms together and extracting common factors from each group. Let's apply this technique to our quartic polynomial: 3x⁴ - 7x³ + 6x - 14.

We can group the first two terms and the last two terms together:

(3x⁴ - 7x³) + (6x - 14)

Now, we extract the greatest common factor (GCF) from each group. From the first group, the GCF is x³, and from the second group, the GCF is 2. Factoring these out, we obtain:

x³(3x - 7) + 2(3x - 7)

Aha! Notice that we now have a common binomial factor of (3x - 7) in both terms. This is a crucial step in the factoring by grouping process. We can factor out this common binomial factor:

(3x - 7)(x³ + 2)

The Completely Factored Form

We've made significant progress! We've successfully factored the quartic polynomial into the product of a linear binomial (3x - 7) and a cubic binomial (x³ + 2). Now, let's revisit our original expression, incorporating the factorization we've achieved:

x(3x⁴ - 7x³ + 6x - 14) = x(3x - 7)(x³ + 2)

At this juncture, we must ask ourselves: Can we factor the cubic binomial (x³ + 2) any further? This is where our knowledge of factoring patterns comes into play. The expression (x³ + 2) resembles the sum of cubes pattern, which has the general form a³ + b³ = (a + b)(a² - ab + b²). However, 2 is not a perfect cube, so we cannot directly apply this pattern.

Therefore, the cubic binomial (x³ + 2) is irreducible over the rational numbers, meaning it cannot be factored further using rational coefficients. Consequently, we have arrived at the completely factored form of the given polynomial:

x(3x - 7)(x³ + 2)

This factored form reveals the polynomial's roots (the values of x that make the polynomial equal to zero) and provides valuable insights into its behavior. The roots can be found by setting each factor equal to zero:

x = 0

3x - 7 = 0 => x = 7/3

x³ + 2 = 0 => x = -∛2 (a real root) and two complex roots

Exploring Alternative Factorizations (and Why They're Not Complete)

It's important to understand why the other options presented are not the completely factored form. Let's examine each one:

Option 1: (x⁴ + 2x)(3x - 7)

This factorization is incorrect because when you expand it, you don't get the original polynomial. To see this, distribute (3x - 7) over (x⁴ + 2x):

(x⁴ + 2x)(3x - 7) = 3x⁵ - 7x⁴ + 6x² - 14x

So, it is actually the original polynomial but it's not completely factored. You can factor out x from the term (x⁴ + 2x), which gives x(x³ + 2)(3x - 7). Therefore, this option represents a partially factored form but not the completely factored form.

Option 2: x⁴(3x - 7)(2x - 1)

This factorization is incorrect. Expanding this expression will not yield the original polynomial. This suggests an error in the factoring process. Factoring requires careful attention to detail and a systematic approach to avoid mistakes.

Option 3: x(3x⁴ - 7x³ + 6x - 14)

This factorization is a good start, as it correctly identifies 'x' as a common factor. However, it stops short of completely factoring the polynomial. The expression within the parentheses, 3x⁴ - 7x³ + 6x - 14, can be further factored using the factoring by grouping technique, as we demonstrated earlier.

Significance of Complete Factorization

The quest for the completely factored form is not merely an algebraic exercise; it has profound implications in various mathematical contexts. A completely factored polynomial reveals critical information about the polynomial's behavior, including:

  • Roots (Zeros): The factored form directly exposes the roots of the polynomial, which are the values of 'x' that make the polynomial equal to zero. These roots are crucial for solving equations, graphing polynomials, and analyzing their behavior.
  • Simplification: Factoring simplifies complex expressions, making them easier to manipulate and analyze. This is particularly useful in calculus, where factored forms can streamline differentiation and integration processes.
  • Divisibility: The factors of a polynomial indicate its divisibility. If (x - a) is a factor of a polynomial, then the polynomial is divisible by (x - a). This property is essential in polynomial division and the Remainder Theorem.
  • Graphing: The roots of a polynomial correspond to the x-intercepts of its graph. The factored form, along with other techniques, aids in sketching the graph of a polynomial function.

In conclusion, the completely factored form of 3x⁵ - 7x⁴ + 6x² - 14x is x(3x - 7)(x³ + 2). This factorization unveils the polynomial's structure, its roots, and its divisibility properties, underscoring the significance of complete factorization in algebra and beyond.

Mastering Factoring Techniques

Factoring polynomials is a skill that requires practice and a solid understanding of various techniques. Here are some key strategies to master:

  • Greatest Common Factor (GCF): Always begin by looking for a GCF that can be factored out from all terms.
  • Factoring by Grouping: This technique is effective for polynomials with four or more terms. Group terms strategically and extract common factors.
  • Special Factoring Patterns: Recognize and apply patterns like the difference of squares (a² - b²), the sum/difference of cubes (a³ ± b³), and perfect square trinomials (a² ± 2ab + b²).
  • Trial and Error: For quadratic trinomials (ax² + bx + c), trial and error can be used to find factors.
  • The Rational Root Theorem: This theorem helps identify potential rational roots of a polynomial, which can aid in factoring.

By diligently practicing these techniques and developing a systematic approach, you can confidently tackle a wide range of factoring problems.

Real-World Applications of Polynomial Factoring

While factoring polynomials may seem like an abstract mathematical concept, it has practical applications in various real-world scenarios:

  • Engineering: Factoring is used in structural engineering to analyze stresses and strains in materials and designs.
  • Physics: Polynomials and their factored forms appear in equations describing projectile motion, oscillations, and other physical phenomena.
  • Computer Graphics: Polynomial curves, which rely on factoring, are used to create smooth shapes and animations in computer graphics.
  • Economics: Polynomial functions can model economic phenomena, and factoring can help analyze market trends and predict outcomes.

By understanding the power of factoring, we can appreciate its role in solving problems across diverse fields.

In essence, the completely factored form is the ultimate destination in our factoring journey, providing a comprehensive understanding of the polynomial's structure and behavior. Mastering factoring techniques not only enhances our algebraic skills but also unlocks a gateway to solving real-world problems in various disciplines.