Given P ( x ) = x 3 − 3 x 2 + 25 x − 75 , find the zeros, real and non-real of P The zeros are Write P in factored form (as a product of linear factors).

Question

Be sure to write the full equation, including P ( x ) =.

Answer 1

P(x) = x^3 -3x^2 +25x - 75

try some simple integers

x P(x)

0 -75

1 -52

2 -43

3 0

x=3 is one zero

x-3 is one factor

use long or synthetic division

divide x-3 into P(x) to get

x^2+25

=(x-5i)(x+5i)

x^2 =-25

x = + or - sqr-25 = +5i, -5i

2 imaginary zeroes: 5i,-5i and 1 real zero: 3

imaginary zeroes always come in conjugate pairs

P(x) =(x-3)(x-5i)(x+5i)