The degree of a polynomial -- the value of the largest exponent -- determines how many roots the polynomial has. If the polynomial is a perfect square, the degree will be even, and all of the roots will be double roots. The degree of the square root of the polynomial will be a polynomial of one-half the degree of the perfect square polynomial.
The graph of a perfect square polynomial will remain above the X axis except for possibly a few places where it dips down to touch the X axis at a tangent point that corresponds to a real multiple root. No matter what sign a value of the square root of a polynomial has, the square of that value will be positive. Because the term with the largest exponent determines the the value of the polynomial at the extreme ends of the X axis, the graph of a perfect square polynomial tends to start high on the left side of the graph, perhaps touching the X axis at a few points, then end high on the right side of the graph.
Complex roots always come in pairs where the roots look like a + bi and a - bi. If a square root polynomial has complex roots, this means that these roots are double. The number of complex roots in the square root polynomial is equal to the degree of that polynomial minus the number of tangent points on the X axis -- each of which represents a real root.
The real roots of a perfect square polynomial are represented by the tangent points that the graphed curve touches the X axis. If r is one of these tangent points then X - r will be a factor of the perfect square polynomial. If a perfect square polynomial is divided by all such factors squared, the quotient polynomial should only have complex roots. For real world problems, only the real roots are considered to be practical -- the complex roots are considered extraneous.