Final answer:
To completely factor the expression 5x^4 − 80, we first factor out the common factor of 5, then recognize the difference of squares in the resultant polynomial, which gives us the completely factored form 5(x^2 + 4)(x + 2)(x − 2).
Step-by-step explanation:
The student has asked to factor completely the polynomial 5x4 − 80. First, notice that 5 is a common factor of both terms. We can factor out the 5, which gives us:
5(x4 − 16)
The expression inside the parentheses is a difference of squares, because 16 is 42. We can factor it as:
5((x2)2 − 42)
This simplifies to:
5((x2 + 4)(x2 − 4))
Further, the term (x2 − 4) is also a difference of squares, which can be factored into (x + 2)(x − 2). Thus, the completely factored form is:
5((x2 + 4)(x + 2)(x − 2))