Question

Finding a closed-form expression for a sum

a. Simplify the following products:
(1 - r)(1+r) =
(1 - r)(1+r+ m^2) =
(1 - r)(1+r+p^2 + y) =
(1 - r)(1+r+ m^2 + p + ... + x) =

Answer 1

The process of simplifying products of binomials such as (1 - r)(1+r) involves the distributive property, and results in the simplified form 1 - r². This method is used to simplify each pair of binomials given in the question.

Step-by-step explanation:

The question raises a common task in algebra, which involves simplifying products of binomials. To simplify the given expressions, we apply the distributive property where each term in the first polynomial is multiplied by each term in the second polynomial. We will provide a step-by-step simplification for each of the products given in the question.

For the first product, (1 - r)(1+r), we apply the distributive property:

• 1 × 1 = 1
• 1 × r = r
• -r × 1 = -r
• -r × r = -r²

Combining these, we get 1 - r², which is a difference of squares.

The remaining expressions become more complex as additional terms are included, but the process of simplification is the same. For example, (1 - r)(1+r+ m²) would be expanded similarly and simplified accordingly. The last expression provided hints at a series expansion, and without specific terms, we can't provide a closed-form expression. In general, the expansion is done by multiplying each term in one binomial by each term in the other binomial and then combining like terms.