Factor completely 5x4 − 80.
5(x2 − 4)(x2 + 4)
5(x − 2)(x + 2)(x + 2)(x + 2)
5(x − 2)(x + 2)(x2 − 4)
5(x − 2)(x + 2)(x2 + 4)
5x4 − 80. Rewritten as 5X^4 - 80 = 5(X^4 - 16)
5(X^2 + 4)(X^2 - 4)
5 x⁴ - 80 = 5 ( x⁴ - 16 )
To factor x⁴ - 16 can use difference of squares formula:
a² - b² = ( a - b ) ( a +b )
In this case:
a = x²
because
x⁴ = ( x² )²
b = 4
because
16 = 4²
so
x⁴ - 16 = ( x² )² - 4² = ( x² - 4 ) ( x² + 4 )
To factor x² - 4 again can use difference of squares formula:
a² - b² = ( a - b ) ( a + b )
In this case:
a = x
b = 2
x² - 4 = ( x )² - ( 2 )² = ( x - 2 ) ( x + 2 )
x⁴ - 16 = ( x² - 4 ) ( x² + 4 ) = ( x - 2 ) ( x + 2 ) ( x² + 4 )
5 x⁴ - 80 = 5 ( x⁴ - 16 )
5 x⁴ - 80 = 5 ( x - 2 ) ( x + 2 ) ( x² + 4 )
oops. I didn't finish factoring the X^2 - 4.
To factor the expression 5x^4 - 80 completely, we need to first look for any common factors. In this case, we can take out a common factor of 5 from both terms, giving us:
5(x^4 - 16)
Now, we need to factor the expression x^4 - 16. This is a difference of squares, which can be factored using the formula (a^2 - b^2) = (a + b)(a - b). In our case, a is x^2 and b is 4, so we have:
5((x^2)^2 - 4^2)
= 5(x^2 + 4)(x^2 - 4)
Now, we can further factor the expression x^2 - 4 using the same difference of squares formula:
= 5(x^2 + 4)(x + 2)(x - 2)
Therefore, the expression 5x^4 - 80 can be factored completely as 5(x^2 + 4)(x + 2)(x - 2). Option 3: 5(x − 2)(x + 2)(x^2 − 4) is the correct answer.