Consider the polynomial p(x) = 16x^4 81y^4. Which of the following are equivalent expressions? Select all that are equivalent.
(4x²-9y^2)(4x² - 9y²)
(4x²-9y^2)(4x² + 9y²)
(4x²+9y^2)(2x-3y) (2x-3y)
(4x²+9y²) (2x + 3y) (2x + 3y)
(4x² +9y²) (2x + 3y)(2x - 3y)
Show your work
First, we expand the polynomial p(x) = 16x^4 - 81y^4:
p(x) = (4x^2)^2 - (9y^2)^2
p(x) = (4x^2 + 9y^2)(4x^2 - 9y^2)
Now, we can check which of the given expressions are equivalent:
1. (4x²-9y^2)(4x² - 9y²)
This expression is equivalent to p(x) as shown above.
2. (4x²-9y^2)(4x² + 9y²)
This expression is not equivalent to p(x) because it results in a different sign in the second term when multiplied out.
3. (4x²+9y²)(2x-3y)(2x-3y)
This expression is not equivalent to p(x) because it squares the last term.
4. (4x²+9y²)(2x + 3y)(2x + 3y)
This expression is not equivalent to p(x) because it squares the last term.
5. (4x² +9y²)(2x + 3y)(2x - 3y)
This expression is equivalent to p(x).
Therefore, the equivalent expressions are:
- (4x²-9y²)(4x² - 9y²)
- (4x² +9y²)(2x + 3y)(2x - 3y)