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CLASS 9 MATHS CHAPTER 2 POLYNOMIALS
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Answers
Step-by-step explanation:
Solutions:-
1.
I) Given Polynomial P(x)=x^3+3x^2+3x+1
Divisor = x+1
We know that
Remainder Theorem:-
P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if P(x) is divided by x-a then the remainder is P(a).
Since , x+1 = 0=>x= -1
Now ,
P(x) is divided by (x+1) then the remainder = P(-1)
=> P(-1) = (-1)^3+3(-1)^2+3(-1)+1
=> P(-1)=-1+3(1)-3+1
=> P(-1) = -1+3-3+1
=> P(-1) = 4-4
=> P(-1) = 0
The remainder = 0
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ii) If P(x) is divided by (x-1/2) then the remainder is P(1/2)
Since x-1/2=0=>x = 1/2
=> P(1/2)=(1/2)^3+3(1/2)^2+3(1/2)+1
=> P(1/2)=(1/8)+3(1/4)+(3/2)+1
=> P(1/2)=(1/8)+(3/4)+(3/2)+1
=> P(1/2)=(1+6+12+8)/8
=> P(1/2)=27/8
The remainder = 27/8
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iii) If P(x) is divided by x then the remainder is P(0).
Since x = 0
=> P(0)=(0)^3+3(0)^2+3(0)+1
=> P(0)=0+0+0+1
=> P(0)=1
The remainder=1
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iv) If P(x) is divided by x+π then the remainder is P(-π).
Since x+π=0=>x = -π
=>P(-π)=(-π)^3+3(-π)^2+3(-π)+1
=> P(-π)=-π^3+3π^2-3π+1
The remainder = -π^3+3π^2-3π+1
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v)If P(x) is divided by 5+2x then the remainder is P(-5/2).
Since 5+2x = 0=>x = -5/2
=> P(-5/2)=(-5/2)^3+3(-5/2)^2+3(-5/2)+1
=> P(-5/2)=(-125/8)+3(25/4)-(15/2)+1
=> P(-5/2)=(-125/8)+(75/4)-(15/2)+1
=> P(-5/2)=(-125+150-60+8)/8
=> P(-5/2)=(158-185)/8
=> P(-5/2)= - 27/8
The remainder = -27/8
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2.Given Polynomial P(x)=x^3-ax^2+6x-a
Divisor = x-a
If P(x) is divided by (x-a) then the remainder is P(a)
=> P(a) = a^3-a(a^2)+6a-a
=> P(a)=a^3-a^3+6a-a
=> P(a)=0+6a-a
=> P(a)=6a-a
=> P(a)=5a
The remainder = 5a
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3.Given Polynomial P(x)=3x^3+7x
We know that
Factor Theorem:-
P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if x-a is a factor of P(x) then P(a)=0 vice-versa.
If 7+3x is a factor then P(-7/3) = 0
Since 7+3x=0=>x = -7/3
=> 3(-7/3)^3+7(-7/3)
=> 3(-343/27)-(49/3)
=> (-343/9)-(49/3)
=> (-343-147)/9
=> -490/9
Since P(-7/3)≠0 then 7+3x is not a factor of 3x^3+7x.
Used formulae:-
Remainder Theorem:-
P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if P(x) is divided by x-a then the remainder is P(a).
Factor Theorem:-
P(x) be a polynomial of the degree greater than or equal to 1 and x-a is another linear polynomial if x-a is a factor of P(x) then P(a)=0 vice-versa.