Question

Select the correct answer. what are the zeros of g(x) = x3 6×2 − 9x − 54? a. 1, 2, 27 b. 3, -3, -6 c. -6, 3, 6 d. 2, -1, 18

1. dangkhoa
The zeroes of the given expression, g(x) = x³ + 6x² – 9x – 54 are 3, -3, and -6, making the option B, a right choice.
In the question, we are asked for the zeroes of the expression, g(x) = x³ + 6x² – 9x – 54.
To find the zeroes, we equate the given expression g(x) to 0 and find the values of the variable x, that satisfy the equation then formed.
Equating g(x) to 0, we get:
g(x) = 0,
or, x³ + 6x² – 9x – 54 = 0.
Now, by grouping, we can show this as:
(x³ + 6x²) – (9x + 54) = 0,
Now, by taking common from the groups, we can show this as:
x²(x + 6) – 9(x + 6) = 0,
or, (x² – 9)(x + 6) = 0.
Now, using the formula, a² – b² = (a – b)(a + b), we can show this as:
(x² – 3²)(x + 6) = 0,
or, (x – 3)(x + 3)(x + 6) = 0, which is the required factor form.
Now, by the zero-product rule, we know that:
Either, x – 3 = 0, ⇒ x = 3,
Or, x + 3 = 0, ⇒ x = -3,
Or, x + 6 = 0, ⇒ x = -6.
Thus, the zeroes of the given expression, g(x) = x³ + 6x² – 9x – 54 are 3, -3, and -6, making the option B, a right choice.