Vocabulary, definitions
A polynomial (poly = lots, nomials = names,) is an expression made up of terms. Terms are separated by plus or minus signs, and terms will generally include a variable, where the most common one we use is x. The highest order (power, index) of the variable determines the order of the expression. If we put a y= in front of the expression, then y is defined as a function. (Properly, every x value produces a single y value but several x may produce the same y).
The expression ax + b has two terms ax and b. Only ax has a variable in it and b is a constant (a, b and c are almost always used to represent constants). So ax+b is a linear expression and y=ax+b is a linear function. There are two reasons for calling this linear: if we sketch them they produce straight lines; the highest order of x (x to the power one) is one. When the highest order is not one, the function or expression is non-linear ...
Similarly ax2+bx+c has three terms, has highest order two and is called quadratic because it includes a square. In the same way, y = ax2+bx+c defines y as a quadratic function, often called a parabola (para=like, bola=ball, from the Greek) because the curve is similar to the path of a ball (actually, that requires a to be negative, but they had the general idea).
In the same way, an x³ means it is cubic expression and function, x⁴ quartic and x⁵ quintic: 1/x or x-1 is a reciprocal (with order minus 1) so a/x would make a reciprocal term and y=1/x is called an hyperbola. The words for the expressions are generally from a Latin base and those for the function are (where different) generally from Greek.
To demonstrate that you have the vocabulary clear, classify these examples.
Expression Order No. of terms Expression Name Function name
ax²+bx+c 2 3 quadratic parabola
2x³+ 4x²+3x+2
25x - 23x³
6x²+12x
x³- 54x² - 212
6/x + 17
2x³- 4x² +2 - x4
6 - 16x⁵
Extension: y³ + 5y = x – 23/x is a sort of function; it has two variables x and y and if you choose a value for x we might find a value for y that fits. This is not explicit (meaning y = something only with x as the variable) so it is called implicit. If there are several possible values for y when you pick an x, then technically it is not a function at all. Proper functions have only one y value for each x you choose, although for any y value there might be more than one x value to match. Implicit forms of an equation are sometimes easier to work with, although explicit forms are usually required for answers.
The concept of a term becomes stretched when we allow brackets in expressions. Consider the equation y = x-1 + (3x+5)(4x-1); this currently has two terms because there is just one plus (or minus) outside the brackets. The second term, (3x+5)(4x-1), has two factors, (3x+5) and (4x-1). In the lower school, a factor is often explained as a whole number divisor of another number, such as recognising that the factors of twelve are 1,2,3,4,6 and 12.
In algebra the same thinking applies but x is less limited as an idea. If we expand the brackets (see Brackets Notes) then (3x+5)(4x-1)= 12x² + 17x –5 has three terms and we could have written our equation as y = 12x² + 17x –5 + x-1 , showing four terms not two. Thus you see that the number of terms applies to the current form of the expression and is not an absolute value for an expression or a function. Note that I wrote the terms in descending order of order of the power (index) of x. I would use ascending or descending order siimply out of habit, generally putting the biggest term first.
One of the depressing or exciting things about maths (depending on your current viewpoint) is that just when you think you have an idea clear, you find it can be extended to a place where you are no longer clear. Many pupils refer to this as ‘moving the goalposts’. There is no doubt that maths is a cumulative subject – it builds on what has gone before in a way that even a language doesn’t quite match. Clarity of definition and the use of precise language (English in my case) allows exact terminology. It is when we let ourselves be vague that errors—particularly in understanding—occur. So my objectives must include persuading you to think and persuading you to try to be precise with language. Just imagine how hard it is in another language (French, Cantonese,....)
