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Question

Apply the division algorithm to find the quotient and remainder on dividing f(x) by g(x).

f(x)=x33x2+5x3, g(x)=x22  [4 MARKS]


Solution

Concept : 1 Mark
Application : 1 Mark
Calculation : 2 Marks

We have,

f(x)=x33x2+5x3 and g(x)=x22

Here, degree ((f(x))=3 and degree (g(x))=2

Therefore, quotient q(x) is of degree 1 and the remainder r(x) is of degree less than 2.

Let q(x)=ax+b and r(x)=cx+d

Using division algorithm, we have

f(x)=g(x)×q(x)+r(x)

x33x2+5x3=(x22)(ax+b)+(cx+d)

x33x2+5x3=ax3+bx2+(c2a)x2b+d

Equating the coefficients of various powers of x on both sides, we get

1 = a                           [On equating the coefficients of x3]

-3 = b                          [On equating the coefficients of x2]

5 = c - 2a

-3 = - 2b + d

Solving these equations, we get

a = 1, b = -3, c = 7 and d = -9

Quotient q(x)=ax+b=x3 and remainder r(x)=7x9

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