This chapter is going to be all about polynomial
expressions in algebra. These are your friends, you have already met, without
being properly introduced! We will properly introduce them to you, and they are
going to be your friends in whatever mathematical journey you undertake from
= a2 + 2a + 1
Now that’s a polynomial. That does not look very
special, does it? We have seen a lots of algebraic expressions already, so why
to bother about these? There are many reasons why polynomials are interesting
and important in mathematics.
For now, we will just take one example showing
their use. Remember, we studied lots of arithmetic and then came to algebra,
thinking of variables as unknown numbers. Actually we can now get back to
numbers and try to write them in the language of algebra.
Consider a number like 5418. It is actually 5
thousand 4 hundred and eighteen.
Write it as:
5 x 1000 + 4 x 100 + 1 x 10 + 8
which again can be written as:
5x103 +4 x 102 + 1 x 101
Now it should be clear what this is about. This is
of the form 5x3 + 4x2 + x + 8, which is a polynomial. How does writing in this form help?
We always write numbers in decimal system, and hence always x = 10. Then what is the fun? Remember
divisibility rules? Recall that a number is divisible by 3 only if the sum of
its digits is divisible by 3. Now notice that if x divided by 3 gives 1 as remainder, then it is the same for x2, x3, etc. They all give remainder 1 when divided by 3. So
you get each digit multiplied by 1, added together, which is the sum of digits.
If that is divisible by 3, so is the whole number. You can check that the rule
for divisibility by 9, or even divisibility by 2 or 5, can be proved similarly
with great ease.
Our purpose is not to prove divisibility rules but
to show that representing numbers as polynomials shows us many new number
patterns. In fact, many many objects of study, not just numbers, can be
represented as polynomials and then we can learn many things about them.
In algebra we think of x2, 5x2–3,
2x+7 etc as functions of x. We draw
pictures to see how the function varies as x
varies, and this is very helpful to understand
the function. And now, it turns out that a good number of functions that we
encounter in science, engineering, business studies, economics, and of course
in mathematics, all can be approximated by polynomials, if not actually be
represented as polynomials. In fact, approximating functions using polynomials
is a fundamental theme in all of higher mathematics and a large number of
people make a living simply by working on this idea.
Polynomials are extensively used in biology,
computer science, communication systems ... the list goes on. The given
pictures (Fig. 3.1, 3.2 & 3.3) may be repsresented as a quardartic
polynomial. We will not only learn what polynomials are but also how we can use
them like in numbers, we add them, multiply them, divide one by another, etc.
Observe the given figures.
The total area of the above figures is 4x2 + 2xy + y2 , we
call this expression as an algebraic expression. Here for different values of x and y we get different values of areas. Since the sides x and y can have different values, they are called variables. Thus, a variable is a symbol which can
have various numerical values.
Variables are usually denoted by letters such as x, y
, z, etc. In the above algebraic
expression the numbers 4, 2 are called constants.
Hence the constant is a symbol, which has a fixed numeric value.
An algebraic expression is a combination of constants and variables
combined together with the help of the four fundamental signs.
Examples of algebraic expression are
Any real number is a constant. We can
form numerical expressions using constants and the four arithmetical
Examples of constant are 1, 5, –32, 3/7, - √2 , 8.432, 1000000 and so on.
The use of variables and constants together in
expressions give us ways of representing a range of numbers, one for each value
of the variable. For instance, we know the expression 2pr, it
stands for the circumference of a circle of radius r. As we vary r, say,
1cm, 4cm, 9cm etc, we get larger and larger circles of circumference 2p, 8p, 18p etc.
The single expression 2pr its a
short and compact description for the circumference of all these circles. We
can use arithmetical operations to combine algebraic expressions and get a rich
language of functions and numbers. Letters used for representing unknown real
numbers called variables are x, y, a, b
and so on.
Any part of a term that is multiplied
by the remaining part of the term is called the coefficient of the remaining
+ 5x - 24 is an algebraic expression containing three terms. The variable of
this expression is x, coefficient of x2 is 1 and the coefficient of x is 5 and the constant is –24 (not 24).