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We can write a polynomial p(x) in the decreasing or increasing order of the powers of x. This way of writing the polynomial is called the standard form of a polynomial.

**Polynomials**

The following dot diagrams show the sequence of
patterns.

(a) Draw the fifth pattern in the sequence in the
space above.

(b) How many dots of each colour will be there in
the 10th pattern?

White: ____________ Colour : ____________

(c) The number of colour dots in the nth diagram is
given by the expression ____________

(d) Write algebraic expressions for the number of
white dots and for the total number of dots.

*Solution:*

(a) Draw the fifth pattern in the sequence in the
space above.

(b) How many dots of each colour will be there in
the 10th pattern?

White: __19__ Colour : __11__

(c) The number of colour dots in the nth diagram is
given by the expression n+1

(d) Write algebraic expressions for the number of
white dots and for the total number of dots.

An algebraic expression of the form p*(x)*= a_{n}x^{n} + a_{n-1}x^{n-1}
+…….+ a_{2}x^{2} + a_{1} + a_{0} is called Polynomial in one variable x of degree ‘*n*’ where a_{0}, a_{1}, a_{2},...
a_{n} are constants (a_{n} ≠ 0) and n is a whole number.

In general polynomials are denoted by
f*(x)* g*(x)* p(t) q(z) and r*(x)*
and so on.

We can write a polynomial *p(x)* in the decreasing or increasing order of the powers of *x*. This way of writing the polynomial is
called the standard form of a polynomial.

For example

(i) 8x^{4}+ 4x^{3} -7x^{2} -9x
+ 6

(ii) 5 – 3y + 6y^{2} + 4y^{3} – y^{4}

In a polynomial of one variable, the highest power
of the variable is called the **degree**
**o**f the polynomial.

In case of a polynomial of more than one variable,
the sum of the powers of the variables in each term is considered and the **highest sum** so obtained is called the
degree of the polynomial.

This is intended as the most **significant** power of the polynomial. Obviously when we write *x*^{2}+5*x* the value of *x*^{2}
becomes much larger than 5*x* for large
values of *x*. So we could think of *x*^{2} + 5*x* being almost the same as *x*^{2}
for large values of *x*. So the higher
the power, the more it dominates. That is why we use the highest power as
important information about the polynomial and give it a name.

Find the degree of each term for the
following polynomial and also find the degree of the polynomial

6*ab*^{8}
+ 5*a* ^{2} *b*^{3} *c* ^{2} - 7*ab* + 4*b*^{2} *c* + 2

*Solution*

Given 6ab^{8} + 5a^{2}b^{3}c^{2}-7ab+4b^{2}c+2

**Degree of
each of the terms is given below. **

6ab^{8} has degree (1+8) = 9

5a^{2}b^{3}c^{2} has degree
(2+3+2) = 7

7ab has degree (1+1) = 2

4b^{2}c has degree (2+1) = 3

The constant term 2 is always regarded as having
degree Zero.

The degree of the polynomial6ab^{8} + 5a^{2}b^{3}c^{2}-7ab+4b^{2}c+2.

= the
largest exponent in the polynomial

= 9

We have said that coefficients can be any real
numbers. What if the coefficient is zero? Well that term becomes zero, so we
won’t write it. What if all the coefficients are zero? We acknowledge that it
exists and give it a name.

It is the polynomial having all its coefficients to
be Zero.

*g *(*t*) = 0*t *^{4}* *+ 0*t*^{2}* *- 0*t *,* h
*(*p*) = 0*p *^{2}* *- 0*p *+ 0

From the above example we see that we cannot talk
of the degree of the Zero polynomial, since the above two have different
degrees but both are Zero polynomial. So we say that the degree of the Zero
polynomial is not defined

The degree of the Zero polynomial is not defined

Classify the following polynomials
based on number of terms.

Classify the following polynomials
based on their degree.

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9th EM Mathematics : Algebra : Polynomials | Solved Example Problems | Mathematics

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