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1. 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.

   Learn more about the zeroes of an expression at

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