Theory

Multiply Parentheses

Multiplying parenthetical terms is an application of the Distributive Property. Each term of the first expression multiplies each term of the second expression.

(a + b) (c + d + e) ⇓ a (c + d + e) + b (c + d + e)

If needed, the Distributive Property can be used again to find the required product.

(a + b) (c + d + e) ⇓ a (c + d + e) + b (c + d + e) ⇓ a c + a d + a e + b c + b d + b e

A particular example is shown below with polynomials.

(x^{3} + x^{2} + 5) (x^{4} + 2 x + 1) = (x^{3} + x^{2} + 5) (x^{4} + 2 x + 1) = x^{3} (x^{4} + 2 x + 1) + x^{2} (x^{4} + 2 x + 1) + 5 (x^{4} + 2 x + 1) x^{7} + 2 x^{4} + x^{3} + x^{6} + 2 x^{3} + x^{2} + 5 x^{4} + 10 x + 5

Usually, it is possible to simplify the result by combining like terms.

(x^{3} + x^{2} + 5) (x^{4} + 2 x + 1) = (x^{3} + x^{2} + 5) (x^{4} + 2 x + 1) = x^{7} + 2 x^{4} + x^{3} + x^{6} + 2 x^{3} + x^{2} + 5 x^{4} + 10 x + 5 x^{7} + 7 x^{4} + 3 x^{3} + x^{6} + x^{2} + 10 x + 5

Sometimes this type of products are represented using a table of products like the one shown below. This table illustrates how the number of total products depends on the number of terms of the expressions being multiplied.

As it can be seen in the table above, the multiplication of an expression of

n

terms times an expression of

m

terms results in

n \times m

products.

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