What Is The Completely Factored Form Of $f(x) = X^3 - 2x^2 - 5x + 6$?A. $f(x) = (x+2)(x-3)(x+6)$B. $ F ( X ) = ( X + 2 ) ( X − 3 ) ( X − 6 ) F(x) = (x+2)(x-3)(x-6) F ( X ) = ( X + 2 ) ( X − 3 ) ( X − 6 ) [/tex]C. $f(x) = (x-2)(x+3)(x-1)$D. $f(x) = (x+2)(x-3)(x-1)$

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Introduction to Factoring Polynomials

Factoring polynomials is a crucial concept in algebra that involves expressing a polynomial as a product of simpler polynomials. This process is essential in solving equations, finding roots, and understanding the behavior of functions. In this article, we will explore the completely factored form of the given polynomial $f(x) = x^3 - 2x^2 - 5x + 6$.

Understanding the Polynomial

Before we proceed with factoring, let's analyze the given polynomial. The polynomial is a cubic function, meaning it has a degree of 3. The general form of a cubic function is $f(x) = ax^3 + bx^2 + cx + d$, where a, b, c, and d are constants. In this case, the polynomial is $f(x) = x^3 - 2x^2 - 5x + 6$.

Factoring the Polynomial

To factor the polynomial, we need to find the roots or factors of the polynomial. The roots of a polynomial are the values of x that make the polynomial equal to zero. In other words, we need to find the values of x that satisfy the equation $f(x) = 0$. Once we find the roots, we can express the polynomial as a product of linear factors.

Using the Rational Root Theorem

The rational root theorem states that if a rational number p/q is a root of the polynomial, then p must be a factor of the constant term, and q must be a factor of the leading coefficient. In this case, the constant term is 6, and the leading coefficient is 1. Therefore, the possible rational roots are ±1, ±2, ±3, and ±6.

Testing the Possible Roots

We can test each possible root by substituting it into the polynomial and checking if the result is equal to zero. Let's start with the positive roots:

  • x = 1: $f(1) = 1^3 - 2(1)^2 - 5(1) + 6 = 1 - 2 - 5 + 6 = 0$
  • x = 2: $f(2) = 2^3 - 2(2)^2 - 5(2) + 6 = 8 - 8 - 10 + 6 = -4$
  • x = 3: $f(3) = 3^3 - 2(3)^2 - 5(3) + 6 = 27 - 18 - 15 + 6 = 0$
  • x = 6: $f(6) = 6^3 - 2(6)^2 - 5(6) + 6 = 216 - 72 - 30 + 6 = 120$

Finding the Factors

We found that x = 1 and x = 3 are roots of the polynomial. Therefore, we can express the polynomial as a product of linear factors:

f(x)=(x−1)(x−3)(x+2)f(x) = (x - 1)(x - 3)(x + 2)

Conclusion

In this article, we explored the completely factored form of the polynomial $f(x) = x^3 - 2x^2 - 5x + 6$. We used the rational root theorem to find the possible roots and tested each root to find the actual roots. We then expressed the polynomial as a product of linear factors. The completely factored form of the polynomial is $f(x) = (x - 1)(x - 3)(x + 2)$.

Comparison with the Options

Let's compare our result with the given options:

  • A. $f(x) = (x+2)(x-3)(x+6)$: This option is incorrect because the factor (x + 6) is not present in our result.
  • B. $f(x) = (x+2)(x-3)(x-6)$: This option is incorrect because the factor (x - 6) is not present in our result.
  • C. $f(x) = (x-2)(x+3)(x-1)$: This option is incorrect because the factor (x + 3) is not present in our result.
  • D. $f(x) = (x+2)(x-3)(x-1)$: This option is incorrect because the factor (x - 1) is present in our result, but the factor (x + 2) is not in the correct position.

Final Answer

The completely factored form of the polynomial $f(x) = x^3 - 2x^2 - 5x + 6$ is $f(x) = (x - 1)(x - 3)(x + 2)$.

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials. This process is essential in solving equations, finding roots, and understanding the behavior of functions.

Q: Why is factoring important in mathematics?

A: Factoring is crucial in mathematics because it helps us understand the properties of polynomials, find their roots, and solve equations. It is also used in various applications, such as physics, engineering, and computer science.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

  • Greatest Common Factor (GCF) factoring: This involves finding the greatest common factor of the terms in the polynomial and factoring it out.
  • Difference of Squares factoring: This involves factoring the difference of two squares, which is a special case of factoring.
  • Sum and Difference factoring: This involves factoring the sum or difference of two terms.
  • Rational Root Theorem factoring: This involves using the rational root theorem to find the possible roots of the polynomial and factoring it accordingly.

Q: How do I factor a polynomial?

A: To factor a polynomial, you need to follow these steps:

  1. Check for GCF: Check if there is a greatest common factor among the terms in the polynomial. If there is, factor it out.
  2. Look for special factoring patterns: Look for special factoring patterns, such as the difference of squares or sum and difference.
  3. Use the rational root theorem: Use the rational root theorem to find the possible roots of the polynomial.
  4. Test the possible roots: Test the possible roots by substituting them into the polynomial and checking if the result is equal to zero.
  5. Express the polynomial as a product of linear factors: Once you find the roots, express the polynomial as a product of linear factors.

Q: What are some common mistakes to avoid when factoring polynomials?

A: Some common mistakes to avoid when factoring polynomials include:

  • Not checking for GCF: Failing to check for the greatest common factor can lead to incorrect factoring.
  • Not looking for special factoring patterns: Failing to look for special factoring patterns can lead to incorrect factoring.
  • Not using the rational root theorem: Failing to use the rational root theorem can lead to incorrect factoring.
  • Not testing the possible roots: Failing to test the possible roots can lead to incorrect factoring.

Q: How do I know if a polynomial is factorable?

A: A polynomial is factorable if it can be expressed as a product of simpler polynomials. To determine if a polynomial is factorable, you need to check if it has any roots or factors.

Q: What are some real-world applications of factoring polynomials?

A: Factoring polynomials has many real-world applications, including:

  • Physics: Factoring polynomials is used to solve equations in physics, such as the motion of objects under the influence of gravity.
  • Engineering: Factoring polynomials is used to design and optimize systems, such as bridges and buildings.
  • Computer Science: Factoring polynomials is used in computer science to solve problems, such as cryptography and coding theory.

Q: Can you provide examples of factoring polynomials?

A: Yes, here are some examples of factoring polynomials:

  • Example 1: Factor the polynomial $f(x) = x^2 + 5x + 6$.
  • Example 2: Factor the polynomial $f(x) = x^3 - 2x^2 - 5x + 6$.
  • Example 3: Factor the polynomial $f(x) = x^4 + 2x^3 - 7x^2 - 8x + 12$.

Q: How do I practice factoring polynomials?

A: To practice factoring polynomials, you can try the following:

  • Practice problems: Try solving practice problems, such as those found in textbooks or online resources.
  • Real-world applications: Try applying factoring to real-world problems, such as those found in physics, engineering, or computer science.
  • Online resources: Try using online resources, such as video tutorials or interactive tools, to practice factoring polynomials.

Q: What are some common mistakes to avoid when practicing factoring polynomials?

A: Some common mistakes to avoid when practicing factoring polynomials include:

  • Not checking for GCF: Failing to check for the greatest common factor can lead to incorrect factoring.
  • Not looking for special factoring patterns: Failing to look for special factoring patterns can lead to incorrect factoring.
  • Not using the rational root theorem: Failing to use the rational root theorem can lead to incorrect factoring.
  • Not testing the possible roots: Failing to test the possible roots can lead to incorrect factoring.