Final answer:
The factors of the polynomial x⁴−12x²−27=0 are (x²+9)(x²−3), found by factoring by grouping and using the difference of squares approach. The correct option is a) (x²+9)(x² −3)
Step-by-step explanation:
The question asks us to identify the factors of the polynomial x⁴−12x²−27=0. To do this, we'll try factoring by grouping, searching for two sets of binomials that multiply to give the original polynomial.
First, we notice that x⁴ and −12x² are perfect squares, suggesting we might use a difference of squares factoring approach. However, −27 is not a perfect square, so we'll look for factors of −27 that can combine with −12x² to form a product of two binomials.
By inspection, we recognize that:
- (x² − 3)(x² + 9) = x⁴ − 3x² + 9x² − 27
- Which simplifies to: x⁴ + 6x² − 27 = x⁴ − 12x² − 27
Therefore, the correct factors for the given polynomial are (x² + 9)(x² − 3).
The factors of the polynomial x⁴−12x²−27=0 are: (x²+9)(x²-3).
To identify the factors, we can solve the polynomial equation by factoring. By factoring x⁴−12x²−27, we get (x²+9)(x²-3). We know that if a * b = 0, either a = 0 or b = 0. Therefore, both (x²+9) = 0 and (x²-3) = 0 need to be satisfied to make the polynomial equation true.
Hence, the factors of the polynomial are (x²+9)(x²-3).
The correct option is a) (x²+9)(x² −3)