The associative law states that when adding or multiplying terms, you can perform the operation in any order without changing the result. For example, (ab)c = a(bc) and (a + b) + c = a + (b + c). Note that this law only applies when you perform the same operation on all the terms. If you mix operations, such as addition and multiplication, in an expression, you cannot change the order and get the same result. For example, ab + c ≠ a (b + c).
The distributive law states that multiplying a number by a sum is equal to multiplying it by each addend and adding the products. For example, a(b + c) = ab + ac. In other words, you can distribute the multiplier across each term in the parentheses. Once you have distributed, you can then remove the parentheses from the equation and continue working with the terms.
You can work with terms that include a coefficient and variable in the same way you work with individual numbers or variables. For example, even though the expression 2x + 3y + 5 implies both multiplication and addition, you can still apply the associative law. For example, (2x + 3y) + 5 = 2x + (3y + 5). You can do this because each coefficient and variable acts as a single term. Similarly, you can distribute a term across parentheses, as in 5x( 4 + 2x) = 20x + 10x^2.
One common use of the distributive law is the FOIL method. FOIL stands for "First, Outer, Inner and Last," and is a mnemonic that helps students remember how to use the distributive property to multiply two binomials. For example, to simplify the expression (x + 1)(x + 1), you must distribute each term in the first binomial across each term in the second. To begin distributing, multiply the first terms, or x * x to get x^2. Next multiply the outer terms, or x * 1 to get 1x. Thirdly, multiply the inner terms, or 1 * x to get 1x; then the last terms, or 1 * 1. Lastly, add these terms together to get x^2 + 2x + 1.