polynomials class ix

2. WHAT IS A POLYNOMIAL

3. TRINOMIAL – IS THE SUM OF THREE MONOMIALS. IT HAS
THREE UNLIKE TERMS. (TRI IMPLIES THREE).
X2 + 2X + 1, 3X2 – 4X + 10, 2X + 3Y + 2
POLYNOMIAL – IS A MONOMIAL OR THE SUM (+) OR
DIFFERENCE (-) OF ONE OR MORE TERMS.
(POLY IMPLIES MANY).
X2 + 2X, 3X3 + X2 + 5X + 6, 4X + 6Y + 8

4. SUBSTITUTING
x2 – 4x + 1
We need to find what this equals when we put a
number in for x.. Like
x = 3
Everywhere you see an x… stick in a 3!
x2 – 4x + 1
= (3)2 – 4(3) + 1
= 9 – 12 + 1
= -2

5. Monomial – is an expression that is a number, a
variable, or a product of a number and one or more
variables. Consequently, a monomial has no variable
in its denominator. It has one term. (mono implies
one).
13, 3x, -57, x2, 4y2, -2xy, or 520x2y2
(notice: no negative exponents, no fractional
exponents)
Binomial – is the sum of two monomials. It has two
unlike terms (bi implies two).
3x + 1, x2 – 4x, 2x + y, or y – y2

6. 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.

7. 3X4 + 5X2 – 7X + 1
THE POLYNOMIAL ABOVE IS IN STANDARD FORM.
STANDARD FORM OF A POLYNOMIAL - MEANS THAT
THE DEGREES OF ITS MONOMIAL TERMS DECREASE
FROM LEFT TO RIGHT.
Polynomial Degree Name using
Degree
Number of
Terms
Name using
number of
terms
7x + 4 1 Linear 2 Binomial
3x2
+ 2x + 1 2 Quadratic 3 Trinomial
4x3
3 Cubic 1 Monomial
9x4
+ 11x 4 Fourth degree 2 Binomial
5 0 Constant 1 monomial
term
termtermterm

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

9. ALGEBRAIC IDENTITIES
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)

10. Q/A ON POLYNOMIALS
Q.1) Factorize:
(i) 9x2 – 16y2 (ii)x3-x
A.1)(i) (9x2 – 16y2) = (3x)2 – (4y)2
= (3x + 4y)(3x – 4y)
therefore, (9x2-16y2) = (3x + 4y)(3x – 4y)
(ii) (x3-x) = x(x2-1)
= x(x+1)(x-1)
therefore, (x3-x) = x(x + 1)(x-1)

11. QUESTIONS ON REMAINDER
THEOREM
Q.) Find the remainder when the polynomial
f(x) = x4 + 2x3 – 3x2 + x – 1 is divided by (x-2).
A.) x-2 = 0 x=2
By remainder theorem, we know that when f(x) is
divided by (x-2), the remainder is x(2).
Now, f(2) = (24 + 2*23 – 3*22 + 2-1)
= (16 + 16 – 12 + 2 – 1) = 21.
Hence, the required remainder is 21.