great ppt on polynomials

6.  Monomial
 Binomial
 Trinomial

7. Algebraic expression that consists only one
term is called monomial.
Eg. X+2

8. Algebraic expression that consists two
terms is called binomial.
Eg. x2 + 2x +5

9. Algebraic expression that consists
three terms is called trinomial.
Eg. X3 +2x2+4x+5

10. The exponent of the highest degree term in a
polynomial is known as its degree.
For Eg. X3 +2x2+4x+5
In this the highest degree is “3”

11. Types of polynomial on the basis of degree are :
Linear polynomial: A polynomial of degree 1 is called a
linear polynomial.
Quadratic polynomial: A polynomial of degree 2 is called
a quadratic polynomial.
Cubic polynomial : A polynomial of degree 3 is called a
cubic polynomial.
Biquadratic polynomial : A polynomial of degree 4 is
called a biquadratic polynomial.

12. Degree Classify by
degree
Classify by
no. of
terms.
5 0 Constant Monomi
al
2x - 4 1 Linear Binomial
3x2 + x 2 Quadrati
c
Binomial
x3 - 4x2 +
1
3 Cubic Polynomial
s

14. Let f(x) be a polynomial of degree n > 1 and let a be any real number.
When f(x) is divided by (x-a) , then the remainder is f(a).
PROOF Suppose when f(x) is divided by (x-a), the quotient is g(x) and the remainder
is r(x).
Then, degree r(x) < degree (x-a)
degree r(x) < 1 [ therefore, degree (x-a)=1]
degree r(x) = 0
r(x) is constant, equal to r (say)
Thus, when f(x) is divided by (x-a), then the quotient is g9x) and the remainder is r.
Therefore, f(x) = (x-a)*g(x) + r (i)
Putting x=a in (i), we get r = f(a)
Thus, when f(x) is divided by (x-a), then the remainder is f(a).

15. Let f(x) be a polynomial of degree n > 1 and let a be
any real number.
(i) If f(a) = 0 then (x-a) is a factor of f(x).
PROOF let f(a) = 0
On dividing f(x) by 9x-a), let g(x) be the quotient.
Also, by remainder theorem, when f(x) is divided
by (x-a), then the remainder is f(a).
therefore f(x) = (x-a)*g(x) + f(a)
f(x) = (x-a)*g(x) [therefore
f(a)=0(given]
(x-a) is a factor of f(x).

16. Some common identities used to factorize polynomials
(a+b)2=a2+b2+2ab (x+a)(x+b)=x2+(a+b)x+ab(a-b)2=a2+b2-2ab a2-b2=(a+b)(a-b)

17. Advanced identities used to factorize polynomials
(x+y+z)2=x2+y2+z2
+2xy+2yz+2zx
(x-y)3=x3-y3-
3xy(x-y)
(x+y)3=x3+y3+
3xy(x+y)
x3+y3=(x+y) *
(x2+y2-xy) x3-y3=(x+y) *
(x2+y2+xy)

18. 1) Polynomials of degree 1, 2 & 3 are called
linear,
quadric and cubic polynomials
respectively.
2) A quadric polynomial can have at most two
zeros and
a cubic polynomial can have three zeros.
3) If we are given with the sum and product
of zeros we can find the polynomial by the
following formula:
K[x2+(sum of zeros)x+(Product of zeros)]

19. 4) If two zeros of a polynomial are given then we can
find the third zero by the following steps:
 Convert the zeros into factors of the polynomial
 Make a combined factor by multiplying the two
factors
 Now divide the polynomial by the combined factor
 Write the quotient separately
 Do middle term splitting
By this process, we can find the third zero of the
polynomial.

20.  A real number ‘a’ is a zero of a polynomial p(x) if p(a)=0.
In this case, a is also called a root of the equation
p(x)=0.
 Every linear polynomial in one variable has a unique
zero, a non-zero constant polynomial has no zero, and
every real number is a zero of the zero polynomial.