Divide the polynomial p(x) by the polynomial g(x) and find the quotient and the remainder in each of the following 1) p(x) =x³-3x² 5x-3,. g(x)=x²-2?

Question

Answer

Division of Polynomial

Given polynomial:

p(x) = x³-3x²+5x-3

g(x) = x²-2

Step 1: Divide the first term of p(x) by the first term of g(x)

Dividing x³ by x², we get x. So, the first term of the quotient is x

Step 2: Multiply the divisor (g(x)) by the first term of the quotient (x)

x*(x²-2) = x³-2x

Step 3: Subtract the above result from the dividend (p(x))

p(x) - x³ + 2x = -3x²+5x-3

Step 4: Repeat the above three steps with the remaining polynomial (-3x²+5x-3)

Dividing -3x² by x², we get -3. So, the second term of the quotient is -3

-3*(x²-2) = -3x²+6

-3x²+5x-3 - (-3x²+6) = 5x-9

Step 5: Repeat the above three steps with the remaining polynomial (5x-9)

Dividing 5x by x², we get 5x². So, the third term of the quotient is 5x²

5x²*(x²-2) = 5x⁴-10x²

5x-9 - (5x⁴-10x²) = -5x⁴+10x²+5x-9

Step 6: The remainder is -5x⁴+10x²+5x-9

Therefore, the quotient is x-3-5x² and the remainder is -5x⁴+10x²+5x-9