The document defines key terms related to polynomials including: monomial, polynomial, binomial, trinomial, degree of a monomial/polynomial, and standard form. It provides examples of classifying polynomials by degree and number of terms. The document also discusses writing polynomials in standard and factored form, finding the leading coefficient, and relating the zeros of a polynomial to its x-intercepts. Examples are worked through of writing polynomials in standard form and classifying them by degree and number of terms.

1. SUBMITTED TO:- SUBMITTED BY:-
VINOD PATIL KRISHNA GUPTA
SAMYAK JAIN
PRADHUM PATHAK
Class:- 10th B
SESSION: 2013-2014
Subject :-Mathematics

3. Monomial: A number, a variable or the product of a number and
one or more variables.
Polynomial: A monomial or a sum of monomials.
Binomial: A polynomial with exactly two terms.
Trinomial: A polynomial with exactly three terms.
Coefficient: A numerical factor in a term of an algebraic
expression.

4. Degree of a monomial: The sum of the exponents of all of the
variables in the monomial.
Degree of a polynomial in one variable: The largest exponent of
that variable.
Standard form: When the terms of a polynomial are arranged
from the largest exponent to the smallest exponent in decreasing
order.

5. What is the degree of the monomial?
24
5 bx
The degree of a monomial is the sum of the exponents of
the variables in the monomial.
The exponents of each variable are 4 and 2. 4+2 = 6.
The degree of the monomial is 6.
The monomial can be referred to as a sixth degree
monomial.

6. A polynomial is a monomial or the sum of monomials
2
4x 83 3
x 1425 2
xx
Each monomial in a polynomial is a term of the polynomial.
The number factor of a term is called the coefficient.
The coefficient of the first term in a polynomial is the lead
coefficient.
A polynomial with two terms is called a binomial.
A polynomial with three terms is called a trinomial.

7. A real number α is a zero of a
polynomial f(x), if f(α) = 0.
e.g. f(x) = x³ - 6x² +11x -6
f(2) = 2³ -6 X 2² +11 X 2 – 6
= 0 .
Hence 2 is a zero of f(x).
The number of zeroes of the
polynomial is the degree of
the polynomial. Therefore a
quadratic polynomial has 2
zeroes and cubic 3 zeroes.

8. 14x
83 3
x
1425 2
xx
The degree of a polynomial in one variable is the largest
exponent of that variable.
2 A constant has no variable. It is a 0 degree polynomial.
This is a 1st degree polynomial. 1st degree polynomials are linear.
This is a 2nd degree polynomial. 2nd degree
polynomials are quadratic.
This is a 3rd degree polynomial. 3rd degree polynomials are
cubic.

9. Classify the polynomials by degree and number of terms.
Polynomial
a.
b.
c.
d.
5
42x
xx2
3
14 23
xx
Degree
Classify by
degree
Classify by
number of
terms
Zero Constant Monomial
First Linear Binomial
Second Quadratic Binomial
Third Cubic Trinomial

10. To rewrite a polynomial in standard form, rearrange
the terms of the polynomial starting with the largest
degree term and ending with the lowest degree term.
The leading coefficient, the coefficient of the
first term in a polynomial written in standard form,
should be positive.

11. Function Of Polynomial
For it to be a Polynomial Function the
exponent has to be positive and whole.
There are 2 forms. Standard Form
where it’s all multiplied out. Factored
Form where it has parenthesis.

12. Degree- the highest power of x when
it’s in Standard Form.
Leading Coefficient- the first
coefficient when the polynomial is in
Standard Form and in order. To find the
leading coefficient you have to put the
polynomial in order by exponent, from
highest to lowest.

13. The x-intercept of the graph and the
zero of its function are opposites.
If 2 of the factors are the same then
there is only one x-intercept. 2 of the
same factors is called multiplicity.
The number of x-intercepts can equal or
be less than the number of factors.
Never more.

14. First degree polynomials have one set of
parenthesis.
Second degree polynomials have two
sets of parenthesis.
Third degree polynomials have three
sets of parenthesis.
Fourth degree polynomials have four
sets of parenthesis.

15. 745 24
xxx
x54
4x 2
x 7
Write the polynomials in standard form.
243
5572 xxxx
3
2x4
x 7x52
5x
)7552(1 234
xxxx
3
2x4
x 7x52
5x
Remember: The lead
coefficient should be
positive in standard
form.
To do this, multiply the
polynomial by –1 using
the distributive
property.

16. Write the polynomials in standard form and identify the
polynomial by degree and number of terms.
23
237 xx1.
2. xx 231 2

17. 23
237 xx
23
237 xx
3
3x 2
2x 7
7231 23
xx
723 23
xx
This is a 3rd degree, or cubic, trinomial.

18. xx 231 2
xx 231 2
2
3x x2 1
This is a 2nd degree, or quadratic, trinomial.