What are the factors of the polynomial f(x) = 5x³ – 24x² – 75x + 14?

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Finding the Factors of the Polynomial

The polynomial in question is f(x) = 5x³ – 24x² – 75x + 14. To determine the factors of this polynomial, we can use several methods including synthetic division, factoring by grouping, or applying the Rational Root Theorem.

Steps to Factor the Polynomial:

  1. Identify Rational Roots: Using the Rational Root Theorem, we can test possible rational roots using the factors of the constant term (14) and the leading coefficient (5). The possible rational roots are ±1, ±2, ±7, ±14, that can be divided by the factors of 5 (±1, ±5).
  2. Test Possible Roots: We substitute these values into the polynomial to check if they yield a result of zero. For example, testing x = 1: 
                   f(1) = 5(1)³ - 24(1)² - 75(1) + 14 = 5 - 24 - 75 + 14 = -80 (not a root)

    Continuing this process, we check other values until we find a root.

  3. Find the Factors: Upon testing, let’s say we find that x = 2 is a root. Using synthetic division with x – 2, we can divide the polynomial: 
                   synthetic division process:
               _____________
               2 |  5   -24   -75   14
                   |      10  -28
                   -------------------
                     5   -14   -75  -14

    This results in a new polynomial: 5x² – 14x – 7.

  4. Factor the Quadratic: Now, we need to factor 5x² – 14x – 7. We look for two numbers that multiply to (5 * -7 = -35) and add to -14. The numbers -17 and 3 work: 5x² – 17x + 3x – 7 = (5x + 1)(x – 7).

Final Factors:

Thus, the complete factorization of the polynomial f(x) = 5x³ – 24x² – 75x + 14 is:

       f(x) = (x - 2)(5x + 1)(x - 7)

Conclusion:

Finding the factors of polynomials like this requires a bit of trial and error coupled with systematic testing of roots. By following these methods, we find valuable insights into the polynomial’s structure, making it easier to analyze and work with in mathematical applications.

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