Question

Factor the expression (q^2r + s^2t)(q^4r^2 - 6q^2rs^2t + s^4t^2).

A) (q^2r + s^2t)(q^2r - s^2t)(q^2r + s^2t)
B) (q^2r - s^2t)(q^2r + s^2t)(q^4r^2 + s^4t^2)
C) (q^2r + s^2t)(q^2r - s^2t)(q^4r^2 - s^4t^2)
D) (q^2r - s^2t)(q^4r^2 + s^4t^2)

Answer 1

The expression (q^2r + s^2t)(q^4r^2 - 6q^2rs^2t + s^4t^2) factors as (q^2r + s^2t)(q^2r - s^2t)(q^2r + s^2t), which corresponds to option A.

Step-by-step explanation:

The student asked to factor the expression (q^2r + s^2t)(q^4r^2 - 6q^2rs^2t + s^4t^2). This can be seen as a product of a binomial and a trinomial. When we look at the trinomial, we can recognize it as a form of a squared binomial, specifically the square of (q^2r - s^2t), because:

• The first term, q^4r^2, is the square of q^2r.
• The last term, s^4t^2, is the square of s^2t.
• The middle term, -6q^2rs^2t, is negative twice the product of q^2r and s^2t.

Thus, the trinomial factors as (q^2r - s^2t)^2. Hence, the entire expression factors as option A: (q^2r + s^2t)(q^2r - s^2t)(q^2r + s^2t).