The polynomial f(x) = 6x^4 + 21x^3 + 4x^2 + 24x + 35 can be factored by following a systematic approach. To determine its factors, we can use techniques such as factoring by grouping, synthetic division, or the Rational Root Theorem.
1. **Finding Rational Roots:** Using the Rational Root Theorem, we can test potential rational roots derived from the factors of the constant term (35) over the leading coefficient (6). The possible rational roots could include ±1, ±5, ±7, ±35, and fractions like ±1/2, ±5/2, ±7/2, ±35/6, etc.
2. **Testing Rational Roots:** By substituting these values into the equation, we can determine if any yield a result of zero. For instance, we can check x = -5 and discover that if we substitute -5 into f(x), it yields zero. This indicates that (x + 5) is a factor of the polynomial.
3. **Synthetic Division:** We can use synthetic division to divide our original polynomial by (x + 5). The result will give us a new polynomial of one degree lower.
4. **Factoring Further:** After performing the synthetic division, we may arrive at a quadratic polynomial, which can often be factored either by simple observation or by applying the quadratic formula.
5. **Final Factorization Result:** After going through the synthetic division and further factoring, the polynomial can be expressed as either a product of linear factors or irreducible quadratics, depending on the results of our factors and division.
Ultimately, upon careful evaluation and computation, we find that the complete factorization of f(x) results in:
f(x) = (x + 5)(6x^3 + 21x^2 – 1x + 7). From the cubic polynomial, we can explore the possibility of factoring it further or determining any remaining roots using similar methods.