A variable is an expression that stands for an unknown or unspecified number; we usually use a letter from the English alphabet, such as a, x, or V. (Uppercase and lowercase letters count as different variables.) An algebraic expression is an expression built (in a meaningful way) out of constants, variables, and arithmetic operations, such as x + 5, xy2, and 8/p. A variable by itself also counts as an algebraic expression, as does an arithmetic expression with no variables. To evaluate an algebraic expression, you need to be given values of each of the variables in the expression; for example, if you evaluate x + 5 when x is 7, then the result is 7 + 5 = 12, but if you evaluate the same expression when x is 9, then the result is 9 + 5 = 14.
Two expressions are equivalent if they always evaluate to the same result whenever you use the same values for all of the variables; for example, x + 5 and 5 + x are equivalent. To simplify an algebraic expression, you find an equivalent expression in some standard form; for example, x + 5 is considered standard while 5 + x is not, so 5 + x simplifies to x + 5. Exactly what counts as the standard form of an expression will depend on the context that you're working in, so ‘simplify’ doesn't always mean to make things simpler (but it usually at least doesn't make them any more complicated).
There are many different types of algebraic expressions, but most of the expressions in this class are a particular type: a polynomial. A polynomial is an algebraic expression built out of constants and variables using the operations of addition and multiplication, or anything equivalent to this. Thus, anything built using addition, subtraction, opposites, multiplication, division by nonzero constants, and raising to powers of whole constants is a polynomial. (This is because subtraction is adding an opposite, you can take an opposite by multiplying by −1, dividing by a nonzero constant is multiplying by its reciprocal, raising to the power of zero is equivalent to the constant 1, and raising to the power of a nonzero whole number is equivalent to repeated multiplication.) However, it is forbidden in a polynomial to divide by zero or a variable expression, to raise to the power of negative or fractional number or a variable, or to take an absolute value.
Usually the first step in dealing with any polynomial is to simplify it to standard form.
Each term of a polynomial is a monomial; a monomial is any expression built out of constants and variables using only the operation of multiplication, or anything equivalent to this. Anything built using opposites, multiplication, division by nonzero constants, and raising to powers of whole constants is a monomial, but addition or subtraction is forbidden.
Every monomial is a product of factors; you multiply the factors to get the entire term. The first factor is the coefficient, which is always a constant. Then there is one factor for each variable, which consists of that variable raised to the power of a whole constant. This whole constant is the degree of the monomial on that variable. So in the end, a monomial is given by several numbers: the coefficient and the degrees on all of the variables.
Think about these special cases before we look at some examples:
Now here are some examples:
Now if you look back at the first two examples of polynomials, we see that each of these polynomials has three terms, and each of these terms has a coefficient and a degree on x.
So far, I've only talked about degrees on specific variables. The degree of a monomial is the sum of the degrees for the variables in the monomial. For example:
When writing out a polynomial, we always write the terms in order of decreasing degree. The degree of the polynomial as a whole is the largest degree of any of its nonzero terms. Returning to our original examples:
A polynomial whose degree is zero is equivalent to a constant. (Since the constant 0 has no nonzero terms, it technically does not have a degree at all; sometimes people say that its degree is −1, or even −∞, meaning negative infinity.)
A linear expression is a polynomial whose degree is at most 1; in other words, either it is constant or its degree is exactly 1. A linear expression can be built out of constants and variables using only addition and multiplication by constants; any expression built from addition, subtraction, opposites, multiplication by constants, and division by constants is linear, but multiplication by a variable expression or raising to a power (other than 0 or 1) is forbidden.
Similarly, two factors are alike (or like factors) if they are both coefficients or involve the same variable. For example:
At first, we're only going to add polynomials and multiply them by constants; this includes taking their opposites (which is multiplying by the constant −1) and subtracting them (which is adding their opposites). We'll return to multiplying more generally (and also some special cases of division) in the second half of the course.
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