Factoring Polynomials A Step By Step Guide To Factoring 6x³ + 41x² + 80x + 48

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Factoring polynomial functions is a fundamental skill in algebra and calculus. It allows us to understand the behavior of these functions, find their roots (or zeros), and simplify complex expressions. In this comprehensive guide, we will delve into the process of factoring the polynomial function 6x³ + 41x² + 80x + 48. We will explore various techniques, including the Rational Root Theorem, synthetic division, and factoring by grouping. By the end of this guide, you will have a solid understanding of how to factor this polynomial and similar ones.

Understanding Polynomial Functions

Before we dive into the factoring process, let's briefly review what polynomial functions are. A polynomial function is a function that can be expressed in the form:

f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

where:

  • aₙ, aₙ₋₁, ..., a₁, a₀ are constants called coefficients.
  • n is a non-negative integer called the degree of the polynomial.
  • x is the variable.

In our case, the polynomial function is 6x³ + 41x² + 80x + 48. This is a cubic polynomial (degree 3) because the highest power of x is 3. The coefficients are 6, 41, 80, and 48.

The graph of a polynomial function provides valuable information about its behavior, including its roots, turning points, and end behavior. The roots of a polynomial function are the values of x for which f(x) = 0. These are also the x-intercepts of the graph. Factoring a polynomial allows us to find its roots algebraically.

The Importance of Factoring Polynomials

Factoring polynomials is a crucial skill in mathematics for several reasons:

  1. Finding Roots: The roots of a polynomial are the values of x that make the polynomial equal to zero. These roots are essential for solving equations, graphing functions, and understanding the behavior of the polynomial.
  2. Simplifying Expressions: Factoring can simplify complex polynomial expressions, making them easier to work with in further calculations or manipulations.
  3. Solving Equations: Factoring is a key step in solving polynomial equations. By setting the factored form of the polynomial equal to zero, we can easily find the solutions.
  4. Graphing Functions: The roots of a polynomial tell us where the graph intersects the x-axis. This information, along with the end behavior and turning points, helps us sketch the graph of the polynomial function.
  5. Applications in Science and Engineering: Polynomial functions are used to model various phenomena in science and engineering, such as projectile motion, electrical circuits, and chemical reactions. Factoring polynomials is often necessary to analyze and solve these models.

Techniques for Factoring Polynomials

There are several techniques for factoring polynomials, and the best approach depends on the specific polynomial you are dealing with. Here are some common methods:

  1. Greatest Common Factor (GCF): Look for the greatest common factor that divides all terms of the polynomial. Factor out the GCF to simplify the expression.
  2. Factoring by Grouping: This technique is useful for polynomials with four or more terms. Group the terms in pairs and factor out the GCF from each pair. If the resulting binomial factors are the same, you can factor them out.
  3. Special Factoring Patterns: Recognize and apply special factoring patterns, such as the difference of squares (a² - b² = (a + b)(a - b)), the sum of cubes (a³ + b³ = (a + b)(a² - ab + b²)), and the difference of cubes (a³ - b³ = (a - b)(a² + ab + b²)).
  4. Rational Root Theorem: This theorem helps us find potential rational roots of a polynomial. If a polynomial has integer coefficients, then any rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
  5. Synthetic Division: This is a shortcut method for dividing a polynomial by a linear factor (x - c). If the remainder is zero, then (x - c) is a factor of the polynomial.
  6. Quadratic Formula: For quadratic polynomials (degree 2), the quadratic formula can be used to find the roots.

Factoring 6x³ + 41x² + 80x + 48: A Step-by-Step Approach

Now, let's apply these techniques to factor the polynomial function 6x³ + 41x² + 80x + 48.

1. Check for a Greatest Common Factor (GCF)

First, we look for a GCF that divides all the coefficients: 6, 41, 80, and 48. The only common factor is 1, so there is no GCF to factor out.

2. Apply the Rational Root Theorem

The Rational Root Theorem states that any rational root of the polynomial must be of the form p/q, where p is a factor of the constant term (48) and q is a factor of the leading coefficient (6). Let's list the factors:

  • Factors of 48 (p): ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48
  • Factors of 6 (q): ±1, ±2, ±3, ±6

Now, we list all possible rational roots (p/q):

±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48, ±1/2, ±3/2, ±1/3, ±2/3, ±4/3, ±8/3, ±1/6

This gives us a long list of potential roots. We can use synthetic division or direct substitution to test these roots.

3. Use Synthetic Division to Test Potential Roots

Synthetic division is an efficient way to test if a potential root is actually a root of the polynomial. Let's start by testing -3/2:

-3/2 | 6 41 80 48
      | -9 -48 -48
      ----------------
      6 32 32 0

The remainder is 0, so -3/2 is a root of the polynomial, and (x + 3/2) is a factor. The quotient is 6x² + 32x + 32.

4. Factor the Quadratic Quotient

Now we need to factor the quadratic 6x² + 32x + 32. First, we can factor out a common factor of 2:

2(3x² + 16x + 16)

Now we need to factor the quadratic 3x² + 16x + 16. We are looking for two numbers that multiply to (3)(16) = 48 and add up to 16. These numbers are 4 and 12. We can use factoring by grouping:

3x² + 16x + 16 = 3x² + 4x + 12x + 16

= x(3x + 4) + 4(3x + 4)

= (3x + 4)(x + 4)

5. Write the Complete Factored Form

Putting it all together, the factored form of the polynomial is:

6x³ + 41x² + 80x + 48 = 2(x + 3/2)(3x + 4)(x + 4)

To eliminate the fraction, we can multiply the 2 into the first factor:

(2x + 3)(3x + 4)(x + 4)

Verifying the Factored Form

To verify our factored form, we can multiply the factors back together and see if we get the original polynomial:

(2x + 3)(3x + 4)(x + 4) = (6x² + 8x + 9x + 12)(x + 4)

= (6x² + 17x + 12)(x + 4)

= 6x³ + 24x² + 17x² + 68x + 12x + 48

= 6x³ + 41x² + 80x + 48

This matches our original polynomial, so our factored form is correct.

Graphing the Polynomial

The factored form of the polynomial tells us the roots, which are the x-intercepts of the graph. The roots are:

  • 2x + 3 = 0 => x = -3/2
  • 3x + 4 = 0 => x = -4/3
  • x + 4 = 0 => x = -4

The graph of the polynomial will cross the x-axis at these three points. Knowing the roots and the leading coefficient (6, which is positive) helps us sketch the graph. The graph will start in the bottom-left quadrant, cross the x-axis at -4, turn, cross the x-axis again at -3/2, turn again, and finally cross the x-axis at -4/3, ending in the top-right quadrant.

Conclusion

Factoring the polynomial function 6x³ + 41x² + 80x + 48 involves a combination of techniques, including the Rational Root Theorem, synthetic division, and factoring by grouping. By following a systematic approach, we can break down the polynomial into its linear factors, which provide valuable information about its roots and behavior. This process not only enhances our understanding of polynomial functions but also equips us with essential skills for solving equations and graphing functions. Mastering polynomial factorization is a fundamental step in advanced algebra and calculus, and it opens doors to a deeper understanding of mathematical concepts and their applications in various fields.

Key takeaways from this guide include:

  • The importance of the Rational Root Theorem in identifying potential roots.
  • The efficiency of synthetic division in testing these roots and finding factors.
  • The application of factoring by grouping to quadratic quotients.
  • The significance of the factored form in determining the roots and graphing the polynomial.

By practicing these techniques and understanding the underlying concepts, you can confidently tackle a wide range of polynomial factoring problems. Remember to always look for the greatest common factor first, apply the Rational Root Theorem, use synthetic division to test potential roots, and factor the resulting quotients. With perseverance and practice, you will master the art of factoring polynomials and unlock their power in solving mathematical problems.