Question

Choose an equivalent expression for 123 • 129 • 124 • 122.

Answer 1

To simplify the expression \( 123 \cdot 129 \cdot 124 \cdot 122 \), we can recognize that pairs can be grouped to simplify the multiplication:

1. Rearrange the numbers:
\[
(123 \cdot 129) \cdot (124 \cdot 122)
\]

2. Calculate each pair:
We can use a difference of squares approach for \( 123 \cdot 129 \) and \( 124 \cdot 122 \):

\[
123 \cdot 129 = (126 - 3)(126 + 3) = 126^2 - 3^2 = 15876 - 9 = 15867
\]

\[
124 \cdot 122 = (123 + 1)(123 - 1) = 123^2 - 1^2 = 15129 - 1 = 15128
\]

3. Now, multiply the two results together:
\[
15867 \cdot 15128
\]

Instead, to find a potentially simpler equivalent expression, we can look for factors in the original product:

Notice that:
\[
123 = 126 - 3,
\quad 129 = 126 + 3,
\quad 124 = 126 - 2,
\quad 122 = 126 - 4.
\]
These values center around the number 126.

Expanding the product, we can find an equivalent polynomial expression by using:
\[
a = 126.
\]
This creates:
\[
(a - 3)(a + 3)(a - 2)(a - 4).
\]
This can be rewritten as:
\[
(a^2 - 9)(a^2 - 6a + 8),
\]
and we can multiply:
\[
= a^4 - 6a^3 + 8a^2 - 9a^2 + 54a - 72.
\]
Combining the terms leads us to a polynomial expression.

However, for the simplest equivalent expression representation:
\[
123 \cdot 129 \cdot 124 \cdot 122 = (126^2 - 9)(126^2 - 6 \cdot 126 + 8),
\]
yielding an equivalent polynomial form centered around \( 126 \).

Thus, an equivalent expression for \( 123 \cdot 129 \cdot 124 \cdot 122 \) is:
\[
(126^2 - 9)(126^2 - 6 \cdot 126 + 8).
\]

Calculating directly yields the value:
This expression has equivalently represents the original multiplication of the pairs effectively.