Question

the form of a perfect square trinomial is ( ax^2 + bx + c ), where ( a ) and ( c ) must be ____________.
in a perfect square trinomial, the middle term must be twice the product of the ____________ of ( a ) and ( c ).
when factoring perfect square trinomials, the expression is rewritten as ( (mx + n)^2 ), where ( mx ) and ( n ) represent the ____________ of the terms ( a ) and ( c ), respectively.

Explanation:

1. For a trinomial \(ax^2 + bx + c\) to be a perfect square, \(a\) and \(c\) must be perfect squares (positive non-zero real numbers) so they can form the squared terms of the binomial.
2. The middle term \(b\) is twice the product of the square roots of \(a\) and \(c\), e.g., for \((\sqrt{a}x + \sqrt{c})^2 = ax^2 + 2\sqrt{ac}x + c\).
3. When factoring, \(mx = \sqrt{a}x\) and \(n = \sqrt{c}\), so they represent the square roots of \(a\) and \(c\) respectively.

Answer:

1. perfect squares (positive real numbers)
2. square roots
3. square roots