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KENDRIYA VIDYALAYA KENDRIYA VIDYALAYA DHARANGDHRA,DHARANGDHRA,MILATRY AREAMILATRY AREA
CLASS-9’ACLASS-9’A
NAME-CH.HARI-NARAYAN.NAME-CH.HARI-NARAYAN.
PolynomialsPolynomials
Each term of a polynomial is a product of a constant (coefficient) and one or more variables whose exponents are non-negative integers.
e.g. –6a3, 4x3 + x, 3y4 + 2y2 + 1, 6x2y2 – xy + y
-ve e.g.
1
4,5,4 2
xxa
PolynomialPolynomial• The The graphgraph of a polynomial function of degree 3.In of a polynomial function of degree 3.In
mathematicsmathematics, a , a polynomialpolynomial is an is an expressionexpression of of finitefinite length constructed from length constructed from variablesvariables (also (also called called indeterminatesindeterminates) and ) and constantsconstants, using only , using only the operations of the operations of additionaddition, , subtractionsubtraction, , multiplicationmultiplication, and non-negative , and non-negative integerinteger exponentsexponents. However, the division by a constant is . However, the division by a constant is allowed, because the allowed, because the multiplicative inversemultiplicative inverse of a of a non zero constant is also a constant. For example, non zero constant is also a constant. For example, xx2 − 2 − xx/4 + 7 is a polynomial, but /4 + 7 is a polynomial, but by the variable by the variable xx (4/x), and also because its third term contains (4/x), and also because its third term contains an exponent that is not an integer (3/2). The term an exponent that is not an integer (3/2). The term "polynomial" can also be used as an adjective, for "polynomial" can also be used as an adjective, for quantities that can be expressed as a polynomial quantities that can be expressed as a polynomial of some parameter, as in of some parameter, as in polynomial timepolynomial time,, which which is used in is used in computational complexity theorycomputational complexity theory
3.1 Review on Polynomials
(A) Monomials and Polynomials
A monomial is a an algebraic expression containing one term, which may be a constant, a positive integral power of a variable or a product of powers of variables.
e.g. 4, 2x3 and 3x2y
12
2
1340
2
234
234
xx
xxx
xxxx
xxx
xxx
242
35223
23
12
12
2
xx
xx
x
divisor
dividend
quotient
remainder
The degree of a polynomial is equal to the highest degree of its terms.
The terms of a polynomials are usually written in descending order (i.e. the terms are arranged in descending degree).
Equality of PolynomialsEquality of Polynomials
If two polynomials in x are equal for all values of x, then the two polynomials are identical, and the coefficients of like powers of x in the two polynomials must be equal.
Alternative MethodAlternative Method
When x = 2,
3(2)2 - 5(2) - 5 = [A+3(2)](2-2) + B
12-10-5 = B
B = -3When x = 0,
3(0)2 - 5(0) – 5 = [A+3(0)](0-2) + B
-5 = -2A + B
-5 = -2A – 3
-2 = -2A
A = 1
(B) Remainder Theorem
2839
27128 = 3 x 9 + 1
remainder
quotient
divisor
dividend
Applications of Theorems about Applications of Theorems about PolynomialsPolynomials
(A)(A)Use Factor Theorem to factorize aUse Factor Theorem to factorize a
polynomial of degree polynomial of degree 33 or above or above (1) try to put a = +1, -1, +2, -2, +3, -3, …. one by one into
the polynomial until the function is equal to zero.
(2) as the function is equal to zero, then (x – a) is one of the factors.
(3) divide the polynomial by (x – a) to get the quotient which is the other factor of the polynomial.
(4) factorize the quotient by the method you have learnt in before.
4. X-y=(x +y) (x -y)4. X-y=(x +y) (x -y)
(1) (3x+2) (3x-2)(1) (3x+2) (3x-2)
= (3x) –(2)= (3x) –(2)
=9x -4=9x -4
(7x-5) (7x+5)(7x-5) (7x+5)
=(7x) – (5)=(7x) – (5)
=49x - 25=49x - 25
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