Hari narayan class 9-a
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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.
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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
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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
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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
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12
2
1340
2
234
234
xx
xxx
xxxx
xxx
xxx
242
35223
23
12
12
2
xx
xx
x
divisor
dividend
quotient
remainder
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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).
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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.
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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
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(B) Remainder Theorem
2839
27128 = 3 x 9 + 1
remainder
quotient
divisor
dividend
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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.
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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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Maths Presentation Ppt
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