Answer :
To factor and solve the polynomial f(x) = x⁴ - x³ - 39x² + 109x - 70, one would use techniques like the Rational Root Theorem and polynomial division to find rational roots and reduce the polynomial, aiming to find the x-intercepts of the function.
To factor the polynomial f(x) = x⁴ - x³ - 39x² + 109x - 70, and solve the equation when f(x) = 0, we would typically look for rational roots using the Rational Root Theorem. This involves listing possible factors of the constant term (70) and the leading coefficient (1), but this can be a lengthy trial-and-error process.
If we succeed in finding a rational root, we can use polynomial division to divide the original polynomial by the binomial (x - r), where r is the root found. The result will be a polynomial of degree 3, which can potentially be factored further using known techniques such as grouping, synthetic division, or applying the Rational Root Theorem again to the reduced polynomial.
Solving the equation f(x) = 0 involves finding the x-intercepts of the polynomial, which correspond to the points where the graph of the function crosses the x-axis. Factoring the polynomial is an essential step in this process, as it can break down the polynomial into easier-to-solve components.
