To determine if there is an equivalent expression to the polynomial [tex]\( x^2 + 8 \)[/tex], we must first check if the polynomial can be factored or simplified further.
1. Checking for common factors:
The polynomial [tex]\( x^2 + 8 \)[/tex] does not have any common factors other than 1.
2. Attempting to factor the polynomial:
The polynomial [tex]\( x^2 + 8 \)[/tex] is in the form of [tex]\( x^2 + c \)[/tex] where [tex]\( c \)[/tex] is a positive number. It is not a difference of squares (e.g., [tex]\( x^2 - 9 = (x + 3)(x - 3) \)[/tex]), and it cannot be expressed as a perfect square trinomial (e.g., [tex]\( x^2 + 4x + 4 = (x + 2)^2 \)[/tex]).
3. Other algebraic methods:
Neither does the polynomial fit into forms that allow factoring via known algebraic identities or formulas.
After checking these methods, we conclude that [tex]\( x^2 + 8 \)[/tex] does not simplify further and cannot be factored into simpler polynomial expressions.
Thus, the polynomial [tex]\( x^2 + 8 \)[/tex] is already in its simplest form and cannot be further simplified or factored. No other equivalent polynomial expression exists for [tex]\( x^2 + 8 \)[/tex].
