Polynomials

A constant is an expression that stands for a particular number, such as 5, 3.2, or −17/2. An arithmetic expression is an expression built (in a meaningful way) out of constants and arithmetic operations (addition, subtraction, etc), such as 5 + 3, 8² or 9 ÷ 0. A constant by itself also counts as an arithmetic expression. To evaluate an arithmetic expression, you work out its value in a standard notation; the previous examples evaluate to 8, 64, and undefined.

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.

Standard forms

Every polynomial is a sum of terms; you add up the terms to get the entire expression. Here are a couple of examples:

The terms of 2x² + 3x + 4 are 2x², 3x, and 4;
The terms of x³ − 2x² + 3 are x³, −2x², and 3.

The last example shows that, when analysing polynomials, you want to think of subtraction as a form of addition. Here, we think of subtracting 2x² as adding −2x².

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:

If there is no constant factor, then the coefficient is 1.
If there is no exponent on a variable, then the degree on that variable is 1.
If a variable does not appear, then the degree on that variable is 0.

Now here are some examples:

The coefficient of 2x² is 2, and its degree on x is 2.
The coefficient of 3x is 3, and its degree on x is 1. (Remember that x¹ = x, so we can think of 3x as 3x¹.)
The coefficient of 4 is 4, and its degree on x is 0. (Remember that x⁰ = 1, multiplication by which does nothing, so we can think of 4 as 4x⁰.)
The coefficient of x³ is 1, and its degree on x is 3. (Remember that multiplication by 1 does nothing, so we can think of x³ as 1x³.)
The coefficient of −2x² is −2, and its degree on x is 2.
The coefficient of 3 is 3, and its degree on x is 0.

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:

The degree of 4x²y³ is 5, since the degree on x is 2 and the degree on y is 3 (and 2 + 3 = 5).
The degree of 4xy³ is 4, since the degree on x is 1 (and 1 + 3 = 4).
The degree of the constant monomial 7 is 0, since there is nothing to add; the degree on every variable is 0.

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:

The degrees of the terms of 2x² + 3x + 4 are 2, 1, and 0, so the degree of this polynomial is 2.
The degree of x³ − 2x² + 3 is 3, since the degrees of its terms are 3, 2, and 0.

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.

Operations

Two terms are alike (or like terms) if they are the same except for their coefficients. For example:

In 2x + 3x, 2x and 3x are alike, since they are both x-terms;
In 2x + 2y, 2x and 2y are not alike, since one is an x-term but the other is a y-term;
In 2x + 2x², 2x and 2x² are also not alike, since one is an x-term but the other is an x²-term.

You can combine like terms into a single term by adding the coefficients; for example:

2x + 3x = 5x.

This also defines addition of polynomials; for example:

To add 2x² + 3x + 4 and x³ − 2x² + 3, we get x³ + 0x² + 3x + 7, which is x³ + 3x + 7.

Similarly, two factors are alike (or like factors) if they are both coefficients or involve the same variable. For example:

In x²x³, x² and x³ are alike, since they are both x-factors;
In x²y², x² and y² are not alike, since one is an x-factor but the other is a y-factor;
In 2x², 2 and x² are also not alike, since one is a coefficient but the other is an x-factor.

You can combine like factors into a single factor by adding the degrees; for example:

x²x³ = x⁵.

This also defines multiplication of monomials; for example:

To multiply 2x²y³ and x³y², we get 2x⁵y⁵.

Finally, to define multiplication of polynomials, we multiply each term by each term. For example:

To multiply 2x² + 3x + 4 and x³ − 2x² + 3, we get:
- (2x²)(x³) + (2x²)(−2x²) + (2x²)(3) + (3x)(x³) + (3x)(−2x²) + (3x)(3) + (4)(x³) + (4)(−2x²) + (4)(3), which is
- 2x⁵ − 4x⁴ + 6x² + 3x⁴ − 6x³ + 9x + 4x³ − 8x² + 12, which is
- 2x⁵ − 4x⁴ + 3x⁴ − 6x³ + 4x³ + 6x² − 8x² + 9x + 12, which is
- 2x⁵ − x⁴ − 2x³ − 2x² + 9x + 12.

This example also shows the importance of putting the terms of a polynomial in order of decreasing degree.

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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