Question

(a) Find the rational zeros and then the other zeros of the polynomial function f(x)=x²-10x³-62x²-78x-27, that is, solve f(x) = 0.

(b) Factor f(x) into linear factors.
(a) Select the correct choice below and fill in any answer box(es) within your choice.
(Use a comma to separate answers as needed. Type an exact answer, using radicals and i as needed.)
OA. The rational zeros are, and the other zeros are
OB. There are only rational zeros and they are
OC. There are no rational zeros. The other zeros are
(b) The factorization of f(x) into linear factors is.
(Type an expression using x as the variable.)

Answer 1

The rational zeros are ±1, ±3, ±9, ±27 and the other zeros can be found through solving a quadratic equation. The factorization of f(x) into linear factors is (x + 3)(x + 1)(x - 1)(x - 9)(x - 27).

Step-by-step explanation:

To find the rational and other zeroes of the given polynomial f(x) = x² - 10x³ - 62x² - 78x - 27, we can use the Rational Root Theorem to find the possible rational zeroes. The possible rational zeroes are the factors of the constant term divided by the factors of the leading coefficient.

The constant term is -27 and the leading coefficient is 1, so the possible rational zeroes are: ±1, ±3, ±9, ±27.

To find the other zeroes, we can use long division or synthetic division to obtain a quadratic equation, and then solve it using the quadratic formula.

The factorization of f(x) into linear factors is (x + 3)(x + 1)(x - 1)(x - 9)(x - 27).