Statements given in the Options (3) and (4) are the correct.
Polynomial function given in the question is,
f(x) = x³ + 5x² + 6x
Check all the options whether they are true or false.
Option (1)
f(x) = 0 at x = 2,
Substitute the value x = 2 in the polynomial.
f(2) = (2)³ + 5(2)² + 6(2)
= 8 + 20 + 12
= 40
False.
Option (2)
(x + 2) is a factor of f(x).
If it's true, f(x) = 0 will be true for x = -2
Substitute x = -2 in the function,
f(-2) = (-2)³ + 5(-2)² + 6(-2)
= -8 + 20 - 12
= -20 + 20
= 0
True.
Option (3)
f(-2) = 0
f(-2) = (-2)³ + 5(-2)² + 6(-2)
= -8 + 20 - 12
= 0
True.
Option (4)
f(x) divided by (x - 2) has a remainder of 0.
x - 2)x³ + 5x² + 6x(x² + 7x
x³ - 2x²
7x² + 6x
7x² - 14x
20x
Here, remainder is (20x).
False.
Therefore, Options (3) and (4) are correct.
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