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The distributive property and terms/coefficients

The distributive property can be written thus: x(y + z) = xy + xz. In other words, the distributive proeprty teaches that when we multiply a term by a sum in parentheses, we first multiply the term by the first term in the sum and then the second term in the sum. We then add the product of x and y to the product of x and z. This is an example of the distributive property.

"Term" is one of those...well...terms we hear used in mathematics a great deal, but few can give a clear, concise definition of the word. A term is what happens when you have a number (called a 'coefficient') which exists either alone, or as a part of a product, specifically with variables attached to it. These variables are themselves raised by powers, though this is not always explicit, as in the case of a term such as 5x. 3 is a term. 3x is also a term. And yes, 3x^4y^9 is also a term! Variables which contain the same exponents are known as "like terms."

It is important to keep in mind that we cannot add or subtract unlike terms. For example 4x^7y^8 - 2x^7y^2 is not something we can mess with. The reason is that the first and second terms are "unlike" one another. But why? Don't the first and second term have the same exponent? Yes, but the variables of both terms must all be raised to the exact same exponent for them to be like terms. If there is any difference between any of the exponents of any of the variables between the two terms, then the two terms are unlike terms, and we cannot add them together or subtract one from the other. If, on the other hand, we had 4x^7y^8 - 2x^7y^8, then we would get the answer 2x^7y^8. Note that only the coefficient is actually altered. The exponents of the variables remain the same. The subtraction (or the addition, for that matter) does not affect them. In any case, it is clear that the latter equation is possible because the equation contains like terms. The exponents of the variables of both terms are identical, and both can therefore be added together or have one subtracted from the other.