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Answer :
We want to determine which of the candidate expressions are equivalent to
[tex]$$25x^4 - 64.$$[/tex]
Let's consider each option step by step.
1. Option 1:
[tex]$$25x^4 + 40x - 40x - 64.$$[/tex]
Notice that [tex]$40x - 40x = 0$[/tex], so the expression simplifies to
[tex]$$25x^4 - 64.$$[/tex]
This is exactly the same as the given expression.
2. Option 2:
[tex]$$25x^4 + 13x - 13x - 64.$$[/tex]
Here also, [tex]$13x - 13x = 0$[/tex], so it simplifies to
[tex]$$25x^4 - 64.$$[/tex]
Again, this is identical to the given expression.
3. Option 3:
[tex]$$(5x^2 + 8)(5x^2 - 8).$$[/tex]
This is a product in the form of a difference of squares:
[tex]$$a^2 - b^2 = (a + b)(a - b),$$[/tex]
where [tex]$a = 5x^2$[/tex] and [tex]$b = 8$[/tex]. Therefore,
[tex]$$(5x^2 + 8)(5x^2 - 8) = (5x^2)^2 - 8^2 = 25x^4 - 64.$$[/tex]
This matches the given expression.
4. Option 4:
[tex]$$(x^2 + 13)(x^2 - 13).$$[/tex]
Again, this is a difference of squares:
[tex]$$x^4 - 13^2 = x^4 - 169.$$[/tex]
Since [tex]$x^4 - 169$[/tex] is not the same as [tex]$25x^4 - 64$[/tex], this option is not equivalent.
5. Option 5:
[tex]$$(5x^2 - 8)^2.$$[/tex]
When expanded, we have
[tex]$$(5x^2 - 8)^2 = (5x^2)^2 - 2\cdot(5x^2)(8) + 8^2 = 25x^4 - 80x^2 + 64.$$[/tex]
Since this expression contains an [tex]$x^2$[/tex] term and a different constant term, it is not equivalent to [tex]$25x^4 - 64$[/tex].
Thus, only Options 1, 2, and 3 are equivalent to the given expression.
The final answer is: Options 1, 2, and 3.
[tex]$$25x^4 - 64.$$[/tex]
Let's consider each option step by step.
1. Option 1:
[tex]$$25x^4 + 40x - 40x - 64.$$[/tex]
Notice that [tex]$40x - 40x = 0$[/tex], so the expression simplifies to
[tex]$$25x^4 - 64.$$[/tex]
This is exactly the same as the given expression.
2. Option 2:
[tex]$$25x^4 + 13x - 13x - 64.$$[/tex]
Here also, [tex]$13x - 13x = 0$[/tex], so it simplifies to
[tex]$$25x^4 - 64.$$[/tex]
Again, this is identical to the given expression.
3. Option 3:
[tex]$$(5x^2 + 8)(5x^2 - 8).$$[/tex]
This is a product in the form of a difference of squares:
[tex]$$a^2 - b^2 = (a + b)(a - b),$$[/tex]
where [tex]$a = 5x^2$[/tex] and [tex]$b = 8$[/tex]. Therefore,
[tex]$$(5x^2 + 8)(5x^2 - 8) = (5x^2)^2 - 8^2 = 25x^4 - 64.$$[/tex]
This matches the given expression.
4. Option 4:
[tex]$$(x^2 + 13)(x^2 - 13).$$[/tex]
Again, this is a difference of squares:
[tex]$$x^4 - 13^2 = x^4 - 169.$$[/tex]
Since [tex]$x^4 - 169$[/tex] is not the same as [tex]$25x^4 - 64$[/tex], this option is not equivalent.
5. Option 5:
[tex]$$(5x^2 - 8)^2.$$[/tex]
When expanded, we have
[tex]$$(5x^2 - 8)^2 = (5x^2)^2 - 2\cdot(5x^2)(8) + 8^2 = 25x^4 - 80x^2 + 64.$$[/tex]
Since this expression contains an [tex]$x^2$[/tex] term and a different constant term, it is not equivalent to [tex]$25x^4 - 64$[/tex].
Thus, only Options 1, 2, and 3 are equivalent to the given expression.
The final answer is: Options 1, 2, and 3.
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