What expression is equivalent to $\left(5 z^2+3 z+2\right)^2$ ?
We are asked to expand the algebraic expression \((5z^2 + 3z + 2)^2\). This is the square of a trinomial. The general formula for the square of a trinomial is:
\[(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ac\]
where \(a\), \(b\), and \(c\) are the individual terms of the trinomial. In our case, we have \(a = 5z^2\), \(b = 3z\), and \(c = 2\).
Now, let's calculate the individual squared terms:
Next, we calculate the double products (cross-terms):
Now, we combine all these terms to get the expanded expression:
\[(5z^2 + 3z + 2)^2 = 25z^4 + 9z^2 + 4 + 30z^3 + 12z + 20z^2\]
To simplify, we group the like terms (the terms with \(z^2\)):
\[25z^4 + 30z^3 + (9z^2 + 20z^2) + 12z + 4\]
\[= 25z^4 + 30z^3 + 29z^2 + 12z + 4\]
Comparing this result to the given options, we see it matches the final option.
Expanding trinomials like \((5z^2 + 3z + 2)^2\) is a fundamental skill in algebra, useful for simplifying complex physics equations or engineering models. For instance, if you are modeling the trajectory of a projectile where the height is defined by a polynomial, squaring that expression is a common step to calculate potential energy or other related physical properties. This algebraic manipulation ensures you can accurately analyze and simplify complex systems in science and technology.