Math Problem Statement

Which factor is a factor of the function f(x) = 2x^3 - x^2 - 13x - 6?

Solution

To solve this problem, we are given the cubic function f(x)=2x3x213x6f(x) = 2x^3 - x^2 - 13x - 6 and need to determine which of the given factors is a root.

The possible factors are:

  1. (x2)(x - 2)
  2. (x+3)(x + 3)
  3. (2x1)(2x - 1)
  4. (x3)(x - 3)

We can use synthetic division or the Remainder Theorem, which states that if f(a)=0f(a) = 0, then (xa)(x - a) is a factor of the polynomial.

Let’s check each option by substituting the values into the function:

  1. Testing x=2x = 2:

    f(2)=2(2)3(2)213(2)6=2(8)4266=164266=20f(2) = 2(2)^3 - (2)^2 - 13(2) - 6 = 2(8) - 4 - 26 - 6 = 16 - 4 - 26 - 6 = -20

    f(2)0f(2) \neq 0, so (x2)(x - 2) is not a factor.

  2. Testing x=3x = -3:

    f(3)=2(3)3(3)213(3)6=2(27)9+396=549+396=30f(-3) = 2(-3)^3 - (-3)^2 - 13(-3) - 6 = 2(-27) - 9 + 39 - 6 = -54 - 9 + 39 - 6 = -30

    f(3)0f(-3) \neq 0, so (x+3)(x + 3) is not a factor.

  3. Testing x=12x = \frac{1}{2} (from 2x1=02x - 1 = 0):

    f(12)=2(12)3(12)213(12)6f\left( \frac{1}{2} \right) = 2\left(\frac{1}{2}\right)^3 - \left(\frac{1}{2}\right)^2 - 13\left(\frac{1}{2}\right) - 6

    f(12)=2(18)141326=14141326=6.5f\left( \frac{1}{2} \right) = 2\left(\frac{1}{8}\right) - \frac{1}{4} - \frac{13}{2} - 6 = \frac{1}{4} - \frac{1}{4} - \frac{13}{2} - 6 = -6.5

    f(12)0f\left( \frac{1}{2} \right) \neq 0, so (2x1)(2x - 1) is not a factor.

  4. Testing x=3x = 3:

    f(3)=2(3)3(3)213(3)6=2(27)9396=549396=0f(3) = 2(3)^3 - (3)^2 - 13(3) - 6 = 2(27) - 9 - 39 - 6 = 54 - 9 - 39 - 6 = 0

    f(3)=0f(3) = 0, so (x3)(x - 3) is a factor of the polynomial.

Conclusion:

The correct factor of the function is (x3)(x - 3).

Would you like more details on this, or have any further questions? Here are some related questions to expand your understanding:

  1. How do you factor cubic polynomials in general?
  2. Can the synthetic division method be used to confirm factors of higher-degree polynomials?
  3. How does the Remainder Theorem help in determining factors?
  4. What would be the full factorization of f(x)f(x)?
  5. How would the graph of this cubic polynomial look?

Tip: The Remainder Theorem is a quick way to check if a value is a root without fully factoring the polynomial.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Cubic Equations
Factorization

Formulas

f(x) = 2x^3 - x^2 - 13x - 6
Synthetic Division
Remainder Theorem

Theorems

Remainder Theorem

Suitable Grade Level

Grades 10-12