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CLASS 9 MATHS CHAPTER 2 POLYNOMIALS
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Answers
Step-by-step explanation:
Solutions :-
i) Given Polynomial P(x) = 3x+1
If x=-1/3 is a zero of P(x) then it satisfies the given polynomial.
=>P(-1/3) = 3(-1/3)+1
=>P(-1/3)= (-3/3)+1
=> P(-1/3)=-1+1
P(-1/3)=0
x= -1/3 is the zero of the polynomial p(x)=3x+1
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ii)Given polynomial P(x)=5x-π
If x= 4/5 is a zero of P(x) then it satisfies the given polynomial.
=> P(4/5)=5(4/5)-π
=>P(4/5)=4-π
Since P(4/5)≠0 then x= 4/5 is not the zero of the polynomial .
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iii) Given Polynomial P(x)=x^2-1
If x= 1 is a zero of P(x) then it satisfies the given polynomial.
=>P(1)=(1)^2-1
=>P(1)=1-1
=>P(1)=0
x=1 is the zero of the polynomial of P(x)=x^2-1
If x= -1 is a zero of P(x) then it satisfies the given polynomial.
=>P(-1)=(-1)^2-1
=>P(-1)=1-1
=>P(-1)=0
x=-1 is the zero of the polynomial of P(x)=x^2-1
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iv) Given Polynomial P(x)=(x+1)(x-2)
If x= -1 is a zero of P(x) then it satisfies the given polynomial.
=>P(-1)=(-1+1)(-1-2)
=>P(-1)=(0)(-3)
=>P(-1)=0
x=-1 is the zero of the polynomial of P(x)=(x+1)(x-2)
If x= 2 is a zero of P(x) then it satisfies the given polynomial.
=>P(2)=(2+1)(2-2)
=>P(2)=(3)(0)
=>P(2)=0
x=2 is the zero of the polynomial of P(x)=(x+1)(x-2)
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v)Given Polynomial P(x)=x^2
If x= 0 is a zero of P(x) then it satisfies the given polynomial.
=>P(0)=(0)^2
=>P(0)=0
=>P(0)=0
x=0 is the zero of the polynomial of P(x)=x^2
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vi)Given Polynomial P(x)=lx+m
If x= -m/l is a zero of P(x) then it satisfies the given polynomial.
=>P(-m/l)=l(-m/l)+m
=>P(-m/l)=(-lm/l)+m
=>P(-m/l)=(-m)+m
=>P(-m/l)=0
x=-m/l is the zero of the polynomial of P(x)=lx+m
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vii)Given Polynomial P(x) = 3x^2-1
If x=-1/√3 is a zero of P(x) then it satisfies the given polynomial.
=>P(-1/√3) = 3(-1/√3)^2-1
=> P(-1/√3)=3(1/3)-1
=> P(-1/√3)=(3/3)-1
=> P(-1/√3)=1-1
=>P(-1/√3)=0
x= -1/√3 is the zero of the polynomial p(x)=3x^2-1
If x=-2/√3 is a zero of P(x) then it satisfies the given polynomial.
=>P(2/√3) = 3(2/√3)^2-1
=> P(2/√3)=3(4/3)-1
=> P(2/√3)=(12/3)-1
=> P(2/√3)=4-1
=>P(2/√3)=3
x= 2/√3 is not the zero of the polynomial p(x)=3x^2-1
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viii)Given Polynomial P(x) = 2x-1
If x=1/2is a zero of P(x) then it satisfies the given polynomial.
=>P(1/2) = 2(1/2)-1
=> P(1/2)=(2/2)-1
=> P(1/2)=1-1
P(1/2)=0
x= 1/2 is the zero of the polynomial p(x)=2x-1
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Answers:-
i)x= -1/3 is the zero of the polynomial p(x)=3x+1
ii)x= 4/5 is not the zero of the polynomial .
iii)x=1 is the zero of the polynomial of P(x)=x^2-1
x=-1 is the zero of the polynomial of P(x)=x^2-1
iv)x=-1 is the zero of the polynomial of P(x)=(x+1)(x-2)
x=2 is the zero of the polynomial of P(x)=(x+1)(x-2)
v)x=0 is the zero of the polynomial of P(x)=x^2
vi)x=-m/l is the zero of the polynomial of P(x)=lx+m
vii)x= -1/√3 is the zero of the polynomial p(x)=3x^2-1
x= 2/√3 is not the zero of the polynomial p(x)=3x^2-1
viii)x= 1/2 is the zero of the polynomial p(x)=2x-1
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Used formulae:-
If a is a zero of the polynomial P(x) then it satisfies the P(x). i.e. P(a) = 0
It is also known as factor theorem .
Factor Theorem:-
P(x) be a polynomial of the degree greater than or equal to 1 and x-a is a factor of P (x) is P(a)=0 vice-versa.