Question

PRE CAL

f(x)=x^3+x^2+4x+4

Solve with Synthetic division and find all zeros of the function

Answer 1

The given polynomial is f(x) = x³ + x² + 4x + 4

Use the Remainder Theorem to identify the first zero.
From the Rational Zeros Theorem, possible zeros are +/- 1, +/- 2, +/- 4.
f(1) = 1 + 1 + 4 + 4 = 10 (not a zero).
f(-1) = -1 + 1 - 4 + 4 = 0 (This is a zero)

Therefore (x+1) is a factor of f(x).
Use Synthetic Division.

-1 | 1 1 4 4
-1 0 -4
----------------
1 0 4 0

There is no remainder, therefore
f(x) = (x+1)(x²+4)
Because x² = -4 yields x = 2i, -2i, there are a pair of conjugate complex zeros.

Answer:
All the zeros are -1, 2i, -2i.