PRE CALCULUS Polynomials and Rational Functions 4

Question 1

Which inequality represents where f(x) = (x − 2)/(x + 3) > 0?

Accepted Answer

x < −3 or x > 2

Answer

The rational expression is positive when both factors share the same sign: both negative (x < −3) or both positive (x > 2).

Question 2

Which polynomial has roots 2 + 3i and its conjugate, with integer coefficients?

Accepted Answer

x² − 4x + 13

Answer

Complex roots come in conjugate pairs; (x − (2+3i))(x − (2−3i)) = (x−2)² + 9 = x² − 4x + 13.

Question 3

What is the domain of f(x) = √(x² − 9)?

Accepted Answer

(−∞, −3] ∪ [3, ∞)

Answer

x² − 9 ≥ 0 requires x ≤ −3 or x ≥ 3, giving domain (−∞, −3] ∪ [3, ∞).

Question 4

Using synthetic division, what is the value of f(−3) for f(x) = x³ + 4x² − 2x + 1?

Accepted Answer

10

Answer

f(−3) = (−27) + 4(9) − 2(−3) + 1 = −27 + 36 + 6 + 1 = 16. Recalculating: −27+36=9, 9+6=15, 15+1=16, so answer is 16.

Question 5

A polynomial f(x) of degree 5 has exactly 3 distinct real zeros. How many non-real complex zeros must it have?

Accepted Answer

2

Answer

Degree 5 means 5 zeros (counting multiplicity); non-real complex zeros come in conjugate pairs, so 5 − 3 = 2 complex zeros.

PRE CALCULUS Practice Test