When adding two numbers, such as 123 and 423, care is taken to first line them up and then add like digits. how does expanding this expression to [(1 × 102) + (2 × 101) + (3 × 100)] + [(4 × 102) + (2 × 101) + (3 × 100)] make the operation more like a polynomial addition problem?

Think of 10^2, 10^1, and 10^0 as x^2, x^1, and x^0.
When you add polynomials, you can only combine like terms.
When you add expanded numbers, you can only combine like powers of 10.

123 + 423 =

= (100 + 20 + 3) + (400 + 20 + 3)

= (1 * 10^2 + 2 * 10^1 + 3 * 10^0) + (4 * 10^2 + 2 * 10^1 + 3 * 10^0)

= (1 * 10^2 + 4 * 10^2) + (2 * 10^1 + 2 * 10^1) + (3 * 10^0 + 3 * 10^0)

= 5 * 10^2 + 4 * 10^1 + 6 * 10^0

= 500 + 40 + 6

= 546

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