What expression is equivalent to $(5 z^2+3 z+2)^2$?
A. $5 z^4+3 z^2+4$
B. $5 z^4+9 z^2+4$
C. $25 z^4+30 z^3+19 z^2+12 z+4$
D. $25 z^4+30 z^3+29 z^2+12 z+4

Expand the expression ( 5 z 2 + 3 z + 2 ) 2 by multiplying it by itself.
Multiply each term in the first polynomial by each term in the second polynomial.
Combine the like terms.
The equivalent expression is 25 z 4 + 30 z 3 + 29 z 2 + 12 z + 4 ​ .

Explanation

Understanding the Problem We are given the expression ( 5 z 2 + 3 z + 2 ) 2 and asked to find an equivalent expression from the options provided.

Expanding the Expression To find the equivalent expression, we need to expand the given expression by multiplying the polynomial by itself: ( 5 z 2 + 3 z + 2 ) 2 = ( 5 z 2 + 3 z + 2 ) ( 5 z 2 + 3 z + 2 )

Performing the Multiplication We multiply each term in the first polynomial by each term in the second polynomial:


( 5 z 2 + 3 z + 2 ) ( 5 z 2 + 3 z + 2 ) = 5 z 2 ( 5 z 2 + 3 z + 2 ) + 3 z ( 5 z 2 + 3 z + 2 ) + 2 ( 5 z 2 + 3 z + 2 ) = ( 25 z 4 + 15 z 3 + 10 z 2 ) + ( 15 z 3 + 9 z 2 + 6 z ) + ( 10 z 2 + 6 z + 4 )

Combining Like Terms Now, we combine like terms: 25 z 4 + ( 15 z 3 + 15 z 3 ) + ( 10 z 2 + 9 z 2 + 10 z 2 ) + ( 6 z + 6 z ) + 4 = 25 z 4 + 30 z 3 + 29 z 2 + 12 z + 4

Final Answer Therefore, the expression equivalent to ( 5 z 2 + 3 z + 2 ) 2 is 25 z 4 + 30 z 3 + 29 z 2 + 12 z + 4 .


Examples
Polynomial expansion, like the one we just did, is used in various fields such as physics, engineering, and computer science. For example, in physics, you might use polynomial expansion to approximate the trajectory of a projectile or to model the behavior of waves. In engineering, it can be used to design filters or control systems. In computer science, polynomial expansion is used in computer graphics and cryptography. Understanding how to expand and simplify polynomial expressions is a fundamental skill that can be applied in many different contexts.

Answered by GinnyAnswer | 2025-07-07