Question

Please help... Find all zeros of f(x) = x^3 - x^2 + x -21 and write a linear factorization of f(x).

Zeros: ?
f(x) = ?

Answer 1

x=3 is a soln...so x-3 is a factor...divide f(x) by (x-3)...u get a quad eqn..solve it...find all roots...hence factorise it...in case the quad eqn returns imaginary roots...u have two factors and 1 zero...else..three factors, 3 zeros

Answer 2

in a polynomial f(x) ax^3 + bx^2 + cx + d

you need to find the factors of a and d

\[a = \pm 1 d = \pm 1, \pm 3, \pm 7.. or .. \pm 21\]

let the factors of d = p and factors of a = q

then a root of the polynomial will be p/q

you just need to find it

so

(x - 3) is a factor because f(3) = 3^3 -3^2 + 3 - 21
f(3) = 0 This is the factor theorem

so if you use polynomial division or synthetic division



you'll find the other factor which will be a quadratic... which will have irrational roots.