Final answer:
Option B) and Option E). The polynomial p(x) = 16x⁴ - 81y⁴ is equivalent to the expressions (4x²-9y²)(4x²+9y²) and (4x²+9y²)(2x+3y)(2x-3y), which are options (b) and (e) respectively.
Step-by-step explanation:
The polynomial given is p(x) = 16x⁴ - 81y⁴. We need to find the equivalent expressions among the options provided by factoring this polynomial. Let's review the choices to determine which ones are equivalent:
- (a) (4x²-9y²)(4x²-9y²): This is the square of a binomial, and it results in a difference of squares, but it won't have a middle term as in the given polynomial.
- (b) (4x²-9y²)(4x²+9y²): This multiplies to a difference of squares, which is equivalent to the given polynomial.
- (c) (4x²+9y²)(2x-3y)(2x-3y): This does not result in the given polynomial, as it contains terms that would not cancel out.
- (d) (4x²+9y²)(2x+3y)(2x+3y): Similar to option c, this is also incorrect for the same reasons.
- (e) (4x²+9y²)(2x+3y)(2x-3y): This expression multiplies out to the original polynomial by first presenting a sum of squares which is then multiplied by a difference of two squares.
Therefore, the equivalent expressions are options (b) and (e), as they represent the factored form of p(x) = 16x⁴ - 81y⁴.