Question

Divide -6x^4+5x^2+1+11x by 2x^2+1 and write down the quotient and the remainder. Also verify your answer

Answer 1

we have to find the quotient and remainder, if -6x⁴ + 5x² + 1 + 11x is divided by 2x² + 1.

solution :

2x² + 1)-6x⁴ + 5x² + 11x + 1(-3x² + 4

-6x⁴ - 3x²

---------------------------

8x² + 11x + 1

8x² + 4

-------------------------------------------

11x - 3

therefore quotient is -3x² + 4 and remainder is 11x - 3.

verification : Euclid lemma , a = bq + r , 0 ≤ r < b

a = -6x⁴ + 5x² + 11x + 1

b = 2x² + 1

q = -3x² + 4

r = 11x - 3

LHS = a = -6x⁴ + 5x² + 11x + 1

RHS = bq + r = (2x² + 1)(-3x² + 4) + (11x - 3)

= -6x⁴ + 8x² - 3x² + 4 + 11x - 3

= -6x⁴ + 5x² + 11x + 1

Hence LHS = RHS

Answer 2

Answer:

solution :

2x² + 1)-6x⁴ + 5x² + 11x + 1(-3x² + 4

-6x⁴ - 3x²

---------------------------

8x² + 11x + 1

8x² + 4

-------------------------------------------

11x - 3

therefore quotient is -3x² + 4 and remainder is 11x - 3.

verification : Euclid lemma , a = bq + r , 0 ≤ r < b

a = -6x⁴ + 5x² + 11x + 1

b = 2x² + 1

q = -3x² + 4

r = 11x - 3

LHS = a = -6x⁴ + 5x² + 11x + 1

RHS = bq + r = (2x² + 1)(-3x² + 4) + (11x - 3)

= -6x⁴ + 8x² - 3x² + 4 + 11x - 3

= -6x⁴ + 5x² + 11x + 1

Hence LHS = RHS