8.5.2. Multiplying a binomial by a trinomial

In this multiplication, we shall have to multiply each of the three terms in the trinomial by each of the two terms in the binomial. We shall get in all 3 × 2 = 6 terms, which may reduce to 5 or less, if the term by term multiplication results in like terms.

When multiplying a binomial and a trinomial, you have to multiply every term in the trinomial by every term in the binomial. For a binomial like (a + 7) and a trinomial like (a^2 + 3a + 5), you first take the first term of the binomial (which is 'a') and multiply it by each term in the trinomial (a^2, 3a, and 5). Then, you do the same with the second term of the binomial (which is '7'). The purpose of this process is to ensure that every combination of terms gets considered, leading to an accurate result. In this case, you end up multiplying 2 terms from the binomial with 3 terms from the trinomial, resulting in a total of 6 products: a × a^2, a × 3a, a × 5, 7 × a^2, 7 × 3a, and 7 × 5.

Think of preparing a meal. If you're making two types of sandwiches (the binomial), say ham and cheese (first type) and peanut butter and jelly (second type), and you want to add three kinds of toppings (the trinomial)—lettuce, tomato, and pickles—to every sandwich. For every type of sandwich, you can add each topping. This means that if you have 2 types of sandwiches and 3 toppings, you'll have a total of 6 different combinations of sandwiches with toppings.

Consider (a+7) × (a^2 +3a +5) =a × (a^2 + 3a + 5) + 7 × (a^2 + 3a + 5)

When you use the distributive law, you distribute each term in the binomial (a and 7) to each term in the trinomial (a^2, 3a, and 5). For instance, when you multiply 'a' with 'a^2', you get 'a^3'. When you multiply 'a' with '3a', you get '3a^2'. Continuing the multiplication, you include '7 × a^2', '7 × 3a', and '7 × 5'. This creates a long expression of terms that may need simplification at the end.

Imagine you're organizing a sports tournament. You have two types of games to organize (say soccer and basketball), and for each game, there are three kinds of awards to distribute (gold, silver, and bronze). Using the distributive law, you reason that each game (type) will have each award category attached to it. So, you will calculate the total number of awards given out across all game types.

Observe, every term in one binomial multiplies every term in the other binomial. = 6a^2 + 9ab + 8ba + 12b^2 = 6a^2 + 17ab + 12b^2 (Since ba = ab)

After you have carried out the multiplication and obtained several products, the next step is to combine like terms to simplify the expression. For example, the products '9ab' and '8ba' are like terms. Since 'ba' and 'ab' are the same, you can combine them into '17ab'. The aim is to express the results in their simplest form.

Think of sorting items in your room. If you've mixed different types of toys and you want to organize them, you first gather all the cars, action figures, and dolls. After that, instead of keeping them mixed, you would combine all the similar toys in one box. This makes it easier to find and manage your toys later.

Thus we get 3 terms as a final result after combination, instead of the 6 initially obtained.

In the multiplication process, although mathematically, we might anticipate having a certain number of distinct products (like 6 terms from 2 terms and 3 terms), the reality is many of those products will share similar variables. After combining like terms, we may end up with fewer distinct terms than we started with.

Continuing the earlier example of organizing toys, this would be similar to how the total number of boxes might be reduced after combining the toys. Instead of six separate boxes for each different toy, you might find you can simply combine them into three or four boxes by grouping similar toys together.

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