Verified Answer :

  • Use the algebraic identity \((x + y)^2 = x^2 + 2xy + y^2\) to expand the expression.
  • Identify \(x = 3a^4\) and \(y = 4\).
  • Compute the squares and the middle term: \((3a^4)^2 = 9a^8\), \(2(3a^4)(4) = 24a^4\), and \(4^2 = 16\).
  • Combine the terms to get the final result: \(\boxed{9a^8 + 24a^4 + 16}\), which corresponds to option d.

Explanation

  1. Analyze the Binomial Expression

We are tasked with expanding the squared binomial expression \((3 a^4 + 4)^2\). To expand a binomial of the form \((x + y)^2\), we use the standard algebraic identity:
\[(x + y)^2 = x^2 + 2xy + y^2\]
In this problem, we identify \(x = 3a^4\) and \(y = 4\). We will now apply this formula by calculating each term individually.

  1. Expand the Binomial using the Identity

First, we calculate the squared terms:

  • The first term is \(x^2 = (3a^4)^2\). Using the power rule \((ab)^n = a^n b^n\) and \((a^m)^n = a^{mn}\), we have:

\[(3a^4)^2 = 3^2 \times (a^4)^2 = 9a^8\]

  • The last term is \(y^2 = 4^2\), which is:

\[4^2 = 16\]
Next, we calculate the middle term, \(2xy\):

  • The middle term is \(2 \times (3a^4) \times 4\). Multiplying these coefficients:

\[2 \times 3 \times 4 = 24\]
So, the middle term is:
\[2 \times (3a^4) \times 4 = 24a^4\]

  1. Combine the Terms and Conclusion

By substituting the calculated terms back into the identity \(x^2 + 2xy + y^2\), we get the expanded expression:
\[9a^8 + 24a^4 + 16\]
Comparing this result to the given options:

A\(6 a^4+8\)
B\(9 a^{16}+24 a^4+16\)
C\(9 a^8+16\)
D\(9 a^8+24 a^4+16\)

Our result matches option (d). Therefore, the correct expansion is \(9 a^8 + 24 a^4 + 16\).

Examples

Expanding binomials is a fundamental skill in algebra with many practical applications. For instance, in structural engineering, when calculating the stress on a beam whose cross-section is represented by a polynomial function, squaring binomials helps determine the total surface area or moment of inertia. Similarly, in computer graphics, polynomial transformations are used to model complex curves. Mastering this technique allows for efficient manipulation and simplification of these mathematical models.

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