P(x) = x^3 + 2x^2 − 3x − 84, c = 4; x − 4 ---Select--- a factor of P(x).

Question

Use the Factor Theorem to show if x − c is a factor of P(x) for the given value of c. P(x) = x^3 + 2x^2 − 3x − 84, c = 4

x − 4  ---Select--- is is not a factor of P(x).

Answer 1

P(4) = 64 + 32 - 12 - 84 = 0, so, (x - 4) is a factor of P(x)

Answer 2

f(x) = x^3 +2x^2 -3x -84

by Descartes' rule of signs

1 change of sign, max 1 + real zero

2 changes of sign when x replaced by -x, max 2- real zeros

either 1 + real and 2 - real zeros

or 1+ real and 2 imaginary zeros

definitely 1 + real

f(0) = -84

f(4) = 4^3 -+2(4^2) -3(4) -84

f(4) = 64+32-12-84 = 0

4 is 1 zero and the only real zero

x-1 is a factor

divide the trinomial by the factor to get

x^2 +6x +21

x = -6/2 +/-(1/2)sqr(36-4(21)

x =-3 +.2isqr3

2 imaginary zeros