What are the possible rational zeros of the polynomial function f(x) = x^4 + 6x^3 + 3x^2 + 17x – 15?

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Finding the Possible Rational Zeros

To determine the possible rational zeros of the polynomial function f(x) = x^4 + 6x^3 + 3x^2 + 17x – 15, we can use the Rational Root Theorem. This theorem states that if a polynomial has a rational solution (p/q), then:

  • p (the numerator) is a factor of the constant term of the polynomial.
  • q (the denominator) is a factor of the leading coefficient of the polynomial.

Identifying the Constant and Leading Coefficient

1. **Identify the constant term**: In this polynomial, the constant term is -15.

2. **Identify the leading coefficient**: The leading coefficient, which is the coefficient of the term with the highest degree (x^4 in this case), is 1.

Finding the Factors

3. **List the factors of the constant term (-15)**:

  • Factors of -15 are: ±1, ±3, ±5, ±15.

4. **List the factors of the leading coefficient (1)**:

  • Factors of 1 are: ±1.

Constructing Possible Rational Zeros

5. **Using the Rational Root Theorem**: Possible rational zeros (p/q) will be the factors of the constant term divided by the factors of the leading coefficient. Since the leading coefficient is ±1, the possible rational zeros are simply the factors of -15.

Thus, the possible rational zeros for the polynomial are:

  • ±1
  • ±3
  • ±5
  • ±15

Conclusion

In summary, the possible rational zeros of the polynomial function f(x) = x^4 + 6x^3 + 3x^2 + 17x – 15 are:

  • 1, -1, 3, -3, 5, -5, 15, -15

These values can be tested further using synthetic division or substitution to find actual zeros of the polynomial.

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