Question

when f (x)=x^4-2x^3+3x^2-ax is divided by x+1 and x-1,we get remainders as 19 and 5 respectively. Find the remainder if f(x) is divided by x-3​

Answer 1

Answer:

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Answer 2

Answer:

Step-by-step explanation

Step1: The given expression f(x)=x^4-2x^3+3x^2-ax+b

for case1 f(x) is divided by x+1 leaves a reminder 19, so substitute x= -1 in the given expression

i.e f(-1)=(-1)^4-2(-1)^3+3(-1)^2-a(-1)+b

=>1+2+3+a+b=19

=>6+a+b=19

=>a+b=19-6

=>a+b=13-------------------------------->(1)

Step 2: The given expression f(x)=x^4-2x^3+3x^2-ax+b

for case 2 f(x) is divided by x-1 leaves a reminder 5, so substitute x= 1 in the given expression

i.e f(1)=(1)^4-2(1)^3+3(1)^2-a(1)+b

=>1-2+3-a+b=5

=>2+b-a=5

=>b-a=5-2

=>b-a=3-------------------------------->(2)

solving equation (1) and (2)

=>a+b=13

=>-a+b=3

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2b=16 =>b=16/2=8

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consider any one of the equation (1) and (2) to find the value of 'a' by substitute 'b'=8

with equation (1)

=>a+b=13

=>a+8=13

=>a=13-8

=>a=5

therefore the value of 'a'=5 and 'b'=8

The given expression f(x)=x^4-2x^3+3x^2-ax+b---------------->(3)

after substituting the value of 'a' and 'b' in the equation 3

we get, f(x)=x^4-2x^3+3x^2-5x+8-------------------------------(4)

step3:

The f(x) is divided by x-3 ,then remainder is, substitute x=3

f(3)=(3)^4-2(3)^3+3(3)^2-5(3)+8

=81-54+27-15+8

f(3)=47

Therefore, f(x) = x^4 – 2x^3 + 3x^2 – ax +b when a=3 and b= 8 is 47