2. POLYNOMIALS
O POLYNOMIAL – A polynomial in one
variable X is an algebraic
expression in X of the form
NOT A POLYNOMIAL – The
expression like 1÷x − 1,∫x+2 etc are
not polynomials .
3. TYPES OF POLYNOMIALS
• Polynomialscontain threetypesof terms:-
O (1) monomial :- A polynomial with oneterm.
O (2) binomial :- A polynomial with two terms.
O (3) trinomial :- A polynomial with threeterms
4. DEGREE OF POLYNOMIAL
O Degree of polynomial- The highest power of x in p(x) is
called the degree of the polynomial p(x).
O EXAMPLE –
1) F(x) = 3x +½ is a polynomial in the variable x of
degree 1.
2) g(y) = 2y² − ⅜ y +7 is a polynomial in the variable y of
degree 2 .
5. ZERO OF A POLYNOMIAL
O A real number k is said to a zero of a
polynomial p(x), if said to be a zero of a
polynomial p(x), if p(k) = 0 . For example,
consider the polynomial p(x) = x³ − 3x − 4
. Then,
O p(−1) = (−1)² − (3(−1) − 4 = 0
O Also, p(4) = (4)² − (3 ×4) − 4 = 0
O Here, − 1 and 4 are called the zeroes of
the quadratic polynomial x² − 3x − 4 .
6. TYPES OF POLYNOMIALS
O Types of polynomials are –
O 1] Constant polynomial
O 2] Linear polynomial
O 3] Quadratic polynomial
O 4] Cubic polynomial
O 5] Bi-quadratic polynomial
7. CONSTANT POLYNOMIAL
O CONSTANT POLYNOMIAL – A polynomial of degree zero
is called a constant polynomial.
O EXAMPLE - F(x) = 7 etc .
O It is also called zero polynomial.
O The degree of the zero polynomial is not defined .
8. LINEAR POLYNOMIAL
O LINEAR POLYNOMIAL – A polynomial of
degree 1 is called a linear polynomial .
O EXAMPLE- 2x−3 , ∫3x +5 etc .
O The most general form of a linear
polynomial is ax + b , a ≠ 0 ,a & b are real.
9. QUADRATIC POLYNOMIAL
OQUADRATIC POLYNOMIAL – A polynomial of degree 2 is
called quadratic polynomial .
OEXAMPLE – 2x² + 3x − ⅔ , y² − 2 etc . More generally ,
any quadratic polynomial in x with real coefficient is of
the form ax² + bx + c , where a, b ,c, are real numbers
and a ≠ 0
10. CUBIC POLYNOMIALS
O CUBIC POLYNOMIAL – A polynomial of
degree 3 is called a cubic polynomial .
O EXAMPLE = 2 − x³ , x³, etc .
O The most general form of a cubic
polynomial with coefficients as real
numbers is ax³ + bx² + cx + d , a ,b ,c ,d
are reals .
11. BI QUADRATIC POLYNMIAL
O BI – QUADRATIC
POLYNOMIAL – A fourth
degree polynomial is called a
biquadratic polynomial .
12. QUADRATIC POLYNOMIAL
O For any quadratic polynomial ax² + bx +c, a
≠ 0, the graph of the corresponding equation
y = ax² + bx + c has one of the two shapes
either open upwards or open downward
depending on whether a>0 or a<0 .these
curves are called parabolas .
13. VALUE OF POLYNOMIAL
O If p(x) is a polynomial in x, and if k is
any real constant, then the real
number obtained by replacing x by k in
p(x), is called the value of p(x) at k,
and is denoted by p(k) . For example ,
consider the polynomial p(x) = x² −3x
−4 . Then, putting x= 2 in the
polynomial , we get p(2) = 2² − 3 × 2 −
4 = − 4 . The value − 6 obtained by
replacing x by 2 in x² − 3x − 4 at x = 2 .
Similarly , p(0) is the value of p(x) at x
= 0 , which is − 4 .
14. HOW TO FIND THE ZERO
OF A LINEAR POLYNOMIAL
O In general, if k is a zero of p(x) = ax
+ b, then p(k) = ak + b = 0, k = − b ÷
a . So, the zero of a linear
polynomial ax + b is − b ÷ a = −
( constant term ) ÷ coefficient of x .
Thus, the zero of a linear
polynomial is related to its
coefficients .
15. GEOMETRICAL MEANING OF
THE ZEROES OF A
POLYNOMIAL
O We know that a real number k is a zero of
the polynomial p(x) if p(K) = 0 . But to
understand the importance of finding the
zeroes of a polynomial, first we shall see
the geometrical meaning of –
O 1) Linear polynomial .
O 2) Quadratic polynomial
O 3) Cubic polynomial
16. GEOMETRICAL MEANING OF
LINEAR POLYNOMIAL
O For a linear polynomial ax + b , a ≠ 0, the graph of y
= ax +b is a straight line . Which intersect the x axis
and which intersect the x axis exactly one point (− b
÷ 2 , 0 ) . Therefore the linear polynomial ax + b , a ≠
0 has exactly one zero .
17. GEOMETRICAL MEANING OF
CUBIC POLYNOMIAL
O The zeroes of a cubic polynomial p(x) are the
x coordinates of the points where the graph
of y = p(x) intersect the x – axis . Also , there
are at most 3 zeroes for the cubic
polynomials . In fact, any polynomial of
degree 3 can have at most three zeroes .
18. RELATIONSHIP BETWEEN
ZEROES OF A POLYNOMIAL
For a quadratic polynomial – In general, if α and β are
the zeroes of a quadratic polynomial p(x) = ax² + bx
+ c , a ≠ 0 , then we know that x − α and x− β are
the factors of p(x) . Therefore ,
O ax² + bx + c = k ( x − α) ( x − β ) ,
O Where k is a constant = k[x² − (α + β)x +αβ]
O = kx² − k( α + β ) x + k αβ
O Comparing the coefficients of x² , x and constant
term on both the sides .
O Therefore , sum of zeroes = − b ÷ a
O = − (coefficients of x) ÷ coefficient of x²
O Product of zeroes = c ÷ a = constant term ÷
coefficient of x²
19. RELATIONSHIP BETWEEN ZERO AND
COEFFICIENT OF A CUBIC POLYNOMIAL
O In general, if α , β , Y are the zeroes of a cubic
polynomial ax³ + bx² + cx + d , then
O α+β+Y = − b÷a
O = − ( Coefficient of x² ) ÷ coefficient of x³
O αβ +βY +Yα =c ÷ a
O = coefficient of x ÷ coefficient of x³
O αβY = − d ÷ a
O = − constant term ÷ coefficient of x³
20. DIVISION ALGORITHEM FOR
POLYNOMIALS
O If p(x) and g(x) are any two polynomials with g(x) ≠ 0,
then we can find polynomials q(x) and r(x) such that –
O p(x) = q(x) × g(x) + r(x)
O Where r(x) = 0 or degree of r(x) < degree of g(x) .
O This result is taken as division algorithm for
polynomials .
21. STANDARD FORM
• The Standard Form for writing apolynomial
isto put thetermswith thehighest degree
first.
• Example: Put this in Standard Form: 3x2
-
7 + 4x3
+ x6
O Thehighest degreeis6, so that goesfirst,
then 3, 2 and then theconstant last:
O x6
+ 4x3
+ 3x2
- 7
22. RemainderTheorem
O Let p(x) beany polynomial of degreegreater than
or equal to oneand let abeany real number. If
p(x) isdivided by linear polynomial x-athen the
reminder isp(a).
• Proof :- Let p(x) beany polynomial of degree
greater than or equal to 1. supposethat when p(x)
isdivided by x-a, thequotient isq(x) and the
reminder isr(x), i.g;
p(x) = (x-a) q(x) +r(x)
23. Sincethedegreeof x-ais1 and thedegreeof
r(x) islessthan thedegreeof x-a,the
degreeof r(x) = 0.
Thismeansthat r(x) isaconstant .say r.
So , for every valueof x, r(x) = r.
Therefore, p(x) = (x-a) q(x) + r
In particular, if x = a, thisequation givesus
p(a) =(a-a) q(a) + r
Which provesthetheorem.
24. FactorTheorem
O Let p(x) beapolynomial of degreen >
1 and let abeany real number. If p(a) =
0 then (x-a) isafactor of p(x).
O PROOF:-By thereminder theorem ,
p(x) = (x-a) q(x) + p(a).
25. 1. If p(a) = 0,then p(x) = (x-a) q(x), which
showsthat x-aisafactor of p(x).
2. Sincex-aisafactor of p(x),
p(x) = (x-a) g(x) for samepolynomial
g(x).
In thiscase, p(a) = (a-a) g(a) =0
