4. 2.1 INTRODUCTION
Definition : An algebraic
expression is an expression
built up from constants,
variables and the algebraic
operations(addition, subtraction
multiplication and division)
for example :
x+3, 3y-8, 7x², 5xy+8z-√3z³ etc
5. Consider the expression 5x³ - 4xyz + 8
In this expression 5x³, - 4xyz and 8 are the
terms.
Terms are added to form expressions.
Terms themselves are formed as product of
factors.
In general, any expression containing one or
more terms with non zero coefficients (and
variables with non negative integers as
exponents) is called a polynomial.
6. TYPES OF POLYNOMIAL(NO. OF TERMS)
A polynomial of one term is called a
monomial.
Examples: 2x, 3xyz, -5, ¾ z etc
A polynomial of two terms is called a
binomial.
Examples: 5y-3, 4z³+7, 5xyz –x etc
A polynomial of three terms is called. a
trinomial
Examples:90xz+16x -¼, x-y-7,
2ax +3by –xy etc
7. DEGREE OF A POLYNOMIAL
The highest power of the variable in a polynomial
is the degree of the polynomial.
So, the degree of the polynomial 3x7+ 6x4- 4x +8
is 7 and the degree of the polynomial
5y6 + 9y3 – 10 is 6.
The degree of a non-zero constant polynomial
is zero.
8. TYPES OF POLYNOMIALS (DEGREE)
A polynomial of degree one is called a linear
polynomial.
Examples: 3x-5, 8x+7y-9z, ½ x-6z-10√2 etc
A polynomial of degree two is called a quadratic
polynomial.
Examples: 3x²-5x+4, 8xy+7y-9z, ½ x-6z²-Π etc
A polynomial of degree three is called a cubic
polynomial
Examples: x³+13x²-5x+14, 8xy+7y-9z³,
x-6z²-8y³
9. POLYNOMIALS IN ONE VARIABLE
A polynomial p(x) in one variable x is an
algebraic expression in x of the form
p(x) = anxn + an-1xn-1 + . . . + a2x2 + ax + a0,
where a0, a1, a2, . . ., an are constants(real
numbers) and an ≠ 0.
a0, a1, a2, . . ., an are respectively the coefficients
of x0, x, x2, . . ., xn, and n is called the degree of
the polynomial. Each of anxn , an-1xn-1 , . . . ,a2x2 ,
ax , a0is called a term of the polynomial p(x).
10. In particular,
if a0= a1 = a2 = . . . = an = 0
(all the constants are zero), we get
the zero polynomial, which is
denoted by 0.
The degree of the zero
polynomial is not defined.
11. ZERO OF A POLYNOMIAL
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.
A quadratic polynomial can have at most 2
zeroes and a cubic polynomial can have at
most 3 zeroes
12. 2.2 Geometrical Meaning of the Zeroes of a
Polynomial
The linear polynomial ax + b, a ≠ 0, has exactly one
zero, namely –b/a the x-coordinate of the point where
the graph of y = ax + b intersects the x-axis.
.
Geometrical Meaning of the Zeroes of a Polynomial
Example : The zero of the linear polynomial
-2x +5 is 5/2 the point where the graph of the
linear equation y = -2x+ 5 meets the x axis.
Please refer the following graph.
14. 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 U either
open upwards or open downwards depending on
whether a > 0 or a < 0.
These curves are called parabolas.
A parabola is a plane curve which is mirror
symmetrical and approximately U-shaped.
Please refer the foll.owing figure
15.
16. The zeroes of a quadratic
polynomial ax² + bx + c, a ≠ 0, are
precisely the x-coordinates of the
points where the parabola
representing y = ax² + bx + c
intersects the x-axis
17. We can see geometrically, from the following graphs,
that a quadratic polynomial can have either two
distinct zeroes or two equal zeroes (i.e., one zero), or no
zero. This also means that a polynomial of degree 2
has at most two zeroes
Fig- 1
18.
19. In general, given a
polynomial p(x) of degree n,
the graph of y = p(x)
intersects the x-axis at at
most n points. Therefore, a
polynomial p(x) of degree n
has at most n zeroes.
20. 2.3 RELATIONSHIP BETWEEN ZEROES AND
COEFFICIENTS OF A POLYNOMIAL
Consider a quadratic polynomial,
say p(x) = 2x² – 7x + 6.
Factorise quadratic polynomials by splitting the
middle term.
2x²– 7x + 6 = 2x² – 4x – 3x + 6 = 2x(x – 2) –3(x – 2)
= (x – 2)(2x – 3)
= (x – 2)(2x – 3)
So, the value of p(x) = 2x² – x + 6 is zero
when x –2 = 0 or 2 x – 3 = 0, i.e., when
x = 2or x = 3/2. So, the zeroes of 2x² – 7x + 6 are
2 and 3/2.
21. Observe that :
Sum of its zeroes = 2 +3/2 =7/2
= − (coefficient of x)/(coefficient ofx²)
Product of its zeroes = 2 X3/2 = 3 = 6/2
= (Constant term)/(coefficient of x²)
22. In general, if α and β are the zeroes of the quadratic
polynomial
p(x) = ax² + bx + c, a≠0 then x – α and x –β are the
factors of p(x). Therefore,
ax² + bx + c = k(x –α ) (x –β ), where k is a constant
= k[x² – (α + β )x + α β
= kx² – k(α + β )x + k α β
Comparing the coefficients of x², x and constant terms on
both the sides, we get
a = k, b = – k(α + β ) and c = kα β
This gives,
α + β = - b/a
αβ = c/a
23. In general, if α and β are the zeroes of
the quadratic polynomial
p(x) = ax² + bx + c, a≠0 then,
α + β = −b/a
αβ = c/a
24. 2.4 DIVISION ALGORITHM FOR
POLYNOMIALS
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
p(x) = g(x) × q(x) + r(x),
where r(x) = 0 or degree of r(x) < degree of g(x).
This result is known as the Division Algorithm
for polynomials
26. So, here the quotient is x² – 6x + 14 and
the remainder is − 31. Also,
(x+2)(x² –6x+ 14) + (−31 )
=x³− 6x² +14x +2x² −12x +28 −31
= x³ −4x² + 2x -3
Therefore,
Dividend = Divisor × Quotient + Remainder
27. 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
p(x) = g(x) × q(x) + r(x),
where r(x) = 0 or degree of r(x) < degree of g(x).
This result is known as the Division
algorithm for polynomials.
28. QUESTIONS FOR PRACTISE:
1)Find the zeros of the polynomial p(x) =3x² - x -4
and verify the relationship between zeros and
coefficients.
2) Divide x³ -3x² +3x + 5 by x -2 and verify
division algorithm for polynomials.
3) Questions from Exercise 2.1,2.2 and 2.3 of
Class X NCERT Mathematics Text book
