when polynomial H(x)=dx³ fx²-7x-6 is divided by x-1 the remainder is -10 and when it is divided by x-3,the remainder is 36.find the values of the constant d and f?

Tifepraise

3 years ago

Using Division Algorithm here:-
Dividend=Divisor×Quotient+Remainder

So, Applying it:−
Let q(x),k(x) be quotient when f(x) is divided by x−1 and x−2 respectively

⇒f(x)=(x−1)q(x)+5
∴f(1)=5 ..... (1)

Also,f(x)=(x−2)k(x)+7
∴f(2)=7 ..... (2)

Now, let ax+b be remainder when f(x) is divided by (x−1)(x−2) and g(x) be quotient.
f(x)=(x−1)(x−2)g(x)+(ax+b)
Using (1) and (2)
5=a+b ...... (3)
7=2a+b ...... (4)

Solving (3) and (4), we get
a=2 and b=3
∴2x+3 is remainder when f(x) is divided by (x−1)(x−2).