The distributive property states that a(b + c) = ab + ac, showing how multiplication distributes over addition.

If the product ab is considered and b is increased by 1, the new product becomes a(b + 1) = ab + a, thus increasing by 'a'.

If both a and b are increased, the product ab increases by a + b + 1.

An identity is an equation that holds true for all values, like (a + b)(a - b) = a^2 - b^2.

Using distributive property, it expands to ab + 8a.

For (a + 1)(b - 1), the expanded form is ab + b - a - 1, showing changes in product.

(a + m)(b + n) = ab + mb + an + mn, indicating total change from the product.

Like terms are terms with the same variable part, allowing addition or subtraction, e.g., 3ab and 5ab are like terms.

3a²(a - b + 1/5) = 3a³ - 3a²b + (3/5)a².

Evaluate and compare the original and modified expressions, such as increasing one and decreasing another.

This expands to a² + 2ab + b², which is the square of a binomial.

Expanding means expressing the product as a sum of terms using the distributive property.

The expansion results in ab + av - ub - uv.

Increasing a by 2 and decreasing b by 2 can leave the product unchanged in certain cases.

The distributive property also holds true with negative integers, e.g., (-x)(y + z) = -xy - xz.

Exponential terms contain variables in the exponent, while polynomial terms are sums of variable products with non-negative integer exponents.

This yields a³ + 3a²b + 3ab² + b³.

Brahmagupta is noted for explicitly stating the distributive property in his work 'Brahmasphuṭasiddhānta'.

A multiplication grid shows products visually represented, helping understand relations between factors.