8.3. Multiplying a Monomial by a Monomial

Expression that contains only one term is called a monomial.

A monomial is an algebraic expression that consists of a single term. This means it contains a coefficient (a numerical factor) multiplied by one or more variables raised to non-negative integer exponents. For example, 3x, -5ab^2, and 7 are all monomials. It is important to recognize that a monomial cannot have addition or subtraction within it; it must be a product.

Think of a monomial like a single item in a shopping cart. Just as you can have an apple (which represents 1 apple) or 3 apples (3x), you can't have a mixture of items (like an apple and a banana together in this case) in the same cart without making it more complex.

We begin with 4 × x =x + x + x + x = 4x as seen earlier. Similarly, 4 × (3x) = 3x + 3x + 3x + 3x = 12x. Now, observe the following products. (i) x × 3y = x × 3 × y = 3 × x × y = 3xy (ii) 5x × 3y = 5 × x × 3 × y = 5 × 3 × x × y = 15xy (iii) 5x × (–3y) = 5 × x × (–3) × y = 5 × (–3) × x × y = –15xy

When multiplying two monomials, you multiply their coefficients (numerical parts) and then combine the variables by adding their exponents. For example, to multiply 4 and 3x, you would multiply the numbers (4 * 3 = 12) and keep the variable x as is, resulting in 12x. In a more complex example like 5x * 3y, you multiply 5 * 3 = 15 and then multiply x * y, resulting in 15xy. If there is a negative coefficient, like in 5x * (-3y), it results in -15xy.

Imagine you’re packing boxes. If one box can contain 5 toys (5x) and you want to pack 3 such boxes (3y), you can think of multiplying the contents of the boxes. So, 5 toys/box * 3 boxes = 15 toys total, which can be represented as 15xy where x could represent toys and y represents boxes.

Some more useful examples follow. Note that 5 × 4 = 20 (iv) 5x × 4x2 =(5 × 4) × (x × x2) = 20 × x3 = 20x3 (v) 5x × (– 4xyz) =(5 × – 4) × (x × xyz) = –20 × (x × x × y × z) = –20x2yz (i.e., algebraic factor of product = algebraic factor of first monomial × algebraic factor of second monomial.)

When you multiply monomials that have the same base variables, you add their exponents. For instance, in 5x * 4x^2, you first multiply the coefficients 5 * 4 = 20 and then for the variable x, you add the exponents. Here it is x^(1+2) = x^3, so the result is 20x^3. Similarly, when multiplying different variables, make sure to keep track of each part to know how it combines, as shown in the example with -20x^2yz.

Think of it this way: if one carton holds 5 apples (5x), and you have 4 cartons stacked, where each carton has an extra space for the same type of apple stacked on top (4x^2), you can multiply those numbers to know how many total apples you can fill in for that space. It’s like arranging in a grid: stack vertically to get more apples in that volume.

Observe the following examples. (i) 2x × 5y × 7z = (2x × 5y) × 7z = 10xy × 7z = 70xyz (ii) 4xy × 5x2y2 × 6x3y3 = (4xy × 5x2y2) × 6x3y3 = 20x3y3 × 6x3y3 = 120x3y3 × x3y3 = 120(x3 × x3)(y3 × y3) = 120x6y6.

When multiplying multiple monomials together, group the multiplications in stages if needed. For example, in 2x * 5y * 7z, you can first multiply 2x and 5y to get 10xy, then multiply that result by 7z to get 70xyz. You can also combine coefficients and the variables separately to ensure easier calculations. Notice that when multiplying monomials with the same variables, you should add their exponents.

Consider a factory where you make toy sets. If one type of toy (2x) can be combined with another type (5y) and then that entire result can be styled or packaged with yet another variant (7z), you can see how the products multiply together to form larger combinations in toy sets, similar to multiplying numbers and keeping track of distinct features.

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