8.4. Multiplying a Monomial by a Polynomial

An expression that contains two terms is called a binomial. An expression containing three terms is a trinomial and so on. In general, an expression containing one or more terms with non-zero coefficients (with variables having non-negative integers as exponents) is called a polynomial.

In mathematics, polynomials are expressions made up of variables and coefficients. A single term is called a monomial, two terms make a binomial, and three terms create a trinomial. For instance, '3x + 4' is a binomial, while '2x^2 + 3x + 5' is a trinomial.

Think of polynomials like a recipe: a monomial is a single ingredient, a binomial combines two ingredients, and a trinomial adds a third ingredient. Just as you can combine various ingredients to create different dishes, you can combine terms in mathematics to create different expressions.

Let us multiply the monomial 3x by the binomial 5y + 2, i.e., find 3x × (5y + 2) = ? Recall that 3x and (5y + 2) represent numbers. Therefore, using the distributive law, 3x × (5y + 2) = (3x × 5y) + (3x × 2) = 15xy + 6x.

Multiplying a monomial by a binomial involves distributing the monomial across each term in the binomial. In our example, we take 3x and multiply it by both 5y and 2 separately. The results of these multiplications (15xy and 6x) are then added together to get the final product of 15xy + 6x.

Imagine you are packaging boxes of fruit. If you have 3 boxes (the monomial) and each box has 5 apples and 2 oranges (the binomial), you would calculate the total number of fruits you have by multiplying the number of boxes by the amount of each type of fruit.

Consider 3p × (4p² + 5p + 7). As in the earlier case, we use the distributive law; 3p × (4p² + 5p + 7) = (3p × 4p²) + (3p × 5p) + (3p × 7) = 12p³ + 15p² + 21p.

When multiplying a monomial by a trinomial, we apply the same distributive law by multiplying the monomial by each term of the trinomial separately. Here, we take 3p and multiply it by 4p², 5p, and 7. This gives us three products, which we then add together to yield the final answer of 12p³ + 15p² + 21p.

Consider a gardener with a set of plants. If the gardener has 3 pots of plants, and each pot has 4 potted flowers, 5 potted herbs, and 7 potted vegetables (this is the trinomial), the gardener can find out the total number of each type by multiplying the number of pots by each type of plant separately.

For example: (–3x) × (–5y + 2) = (–3x) × (–5y) + (–3x) × (2) = 15xy – 6x.

Using the distributive law can also apply to negative numbers. Here, when we multiply –3x by –5y, the product is positive, resulting in 15xy. When we multiply –3x by 2, we get –6x. So, the entire expression simplifies to 15xy – 6x.

If you owe money (represented by –3x) but receive payments (represented by a positive outcome of the binomial) for two different types of loans, you can calculate new totals. The distribution ensures you accurately account for each type of transaction.

What about a binomial × monomial? For example, (5y + 2) × 3x = ? We may use commutative law as: 7 × 3 = 3 × 7

The commutative law states that the order in which two numbers are multiplied does not change the product. So whether we calculate (5y + 2) × 3x or 3x × (5y + 2), we will arrive at the same result, 15xy + 6x. This highlights the flexibility in multiplication – we can rearrange terms as needed.

Imagine a light-weight box filled with 5 bags of flour and two bags of sugar. Whether we multiply the amount of flour first and then add sugar, or combine both types simultaneously, will yield the same total weight of the goods you're carrying.

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