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On dividing x3 – 3x2 + x + 2 by polynomial g(x), the quotient and remainder were x – 2 and 4 – 2x respectively
then g(x) :
then g(x) :
- a)x2 + x + 1
- b)x2 + x – 1
- c)x2 – x – 1
- d)x2 – x + 1
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
On dividing x3– 3x2+ x + 2 by polynomial g(x), the quotient and ...
We have remainder theorem as
x3-3x2+x+2=q(x)*g(x)+r(x)
x3-3x2+x+2=(x-2)g(x)+(4-2x)
g(x)=
So by division method,
g(x)= x2 - x + 1
x3-3x2+x+2=q(x)*g(x)+r(x)
x3-3x2+x+2=(x-2)g(x)+(4-2x)
g(x)=
So by division method,
g(x)= x2 - x + 1
Most Upvoted Answer
On dividing x3– 3x2+ x + 2 by polynomial g(x), the quotient and ...
Explanation:
Given:
- Dividend: x3 – 3x2 + x + 2
- Quotient: x – 2
- Remainder: 4 – 2x
Using the division algorithm:
- Dividend = Quotient × Divisor + Remainder
- x3 – 3x2 + x + 2 = (x – 2) × g(x) + (4 – 2x)
Expanding the right side:
- x3 – 3x2 + x + 2 = xg(x) – 2g(x) + 4 – 2x
Comparing coefficients:
- Coefficient of x3 on the left = Coefficient of x3 on the right (1 = 1)
- Coefficient of x2 on the left = Coefficient of x2 on the right (-3 = -g(x))
- Coefficient of x on the left = Coefficient of x on the right (1 = -2 - 2)
Solving the equations:
- From the second equation: g(x) = 3
- Substituting g(x) = 3 in the third equation: 1 = -2 - 2
- This equation is not possible, so the divisor g(x) must be incorrect
- Trying g(x) = x2 – x + 1
Using the correct divisor:
- x3 – 3x2 + x + 2 = (x – 2)(x2 – x + 1) + (4 – 2x)
Therefore, the correct polynomial g(x) is:
- x2 – x + 1
Therefore, the correct answer is option 'D'.
Given:
- Dividend: x3 – 3x2 + x + 2
- Quotient: x – 2
- Remainder: 4 – 2x
Using the division algorithm:
- Dividend = Quotient × Divisor + Remainder
- x3 – 3x2 + x + 2 = (x – 2) × g(x) + (4 – 2x)
Expanding the right side:
- x3 – 3x2 + x + 2 = xg(x) – 2g(x) + 4 – 2x
Comparing coefficients:
- Coefficient of x3 on the left = Coefficient of x3 on the right (1 = 1)
- Coefficient of x2 on the left = Coefficient of x2 on the right (-3 = -g(x))
- Coefficient of x on the left = Coefficient of x on the right (1 = -2 - 2)
Solving the equations:
- From the second equation: g(x) = 3
- Substituting g(x) = 3 in the third equation: 1 = -2 - 2
- This equation is not possible, so the divisor g(x) must be incorrect
- Trying g(x) = x2 – x + 1
Using the correct divisor:
- x3 – 3x2 + x + 2 = (x – 2)(x2 – x + 1) + (4 – 2x)
Therefore, the correct polynomial g(x) is:
- x2 – x + 1
Therefore, the correct answer is option 'D'.
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Community Answer
On dividing x3– 3x2+ x + 2 by polynomial g(x), the quotient and ...
Look ; Polynomial =divisor(quotient) +remainder
so x^3-3x^2+x+2=divisor(x-2) +(4-2x)
so we get divisor=x^3-3x^2+3x-2/(x-2)
so g(x) =x^2-x+1
so x^3-3x^2+x+2=divisor(x-2) +(4-2x)
so we get divisor=x^3-3x^2+3x-2/(x-2)
so g(x) =x^2-x+1
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On dividing x3– 3x2+ x + 2 by polynomial g(x), the quotient and remainder were x – 2 and 4 – 2x respectivelythen g(x) :a)x2+ x + 1b)x2+ x – 1c)x2– x – 1d)x2– x + 1Correct answer is option 'D'. Can you explain this answer?
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