Question

If on dividing the polynomial `x^4-x^3-13x^2+sx+t` by `(x+3)(x+4)` remainder is `0`, find the value of `s` and `t`?

Answer 1

`s = 121, t = 372`

Explanation:

`frac{x^4 - x^3 - 13x^2 + sx + t}{x^2 + 7x + 12} = x^2 + frac{- 8x^3 - 25x^2 + sx + t}{x^2 + 7x + 12}`

`= x^2 - 8x + frac{(-25+56)x^2 + (96 + s)x + t}{x^2 + 7x + 12}`

`= x^2 - 8x + (31) + frac{(-31*7+96 + s)x + (-31*12 + t)}{x^2 + 7x + 12}`

Answer 2

`s=121` and `t=372`

Explanation:

According to Remainder theorem if `(x-a)` is a factor of the polynomial `f(x)`, then `f(a)=0`.

Now as dividing the polynomial `x^4-x^3-13x^2+sx+t` by `(x+3)(x+4)` results in `0` as remainder, both `(x+3)` and `(x+4)` are a factor of the polynomial `f(x)=x^4-x^3-13x^2+sx+t`

and as per remainder theorem `f(-3)=0` and `f(-4)=0` i.e.

`(-3)^4-(-3)^3-13(-3)^2+s(-3)+t=0`

or `81+27-117-3s+t=0`

i.e. `-3s+t=9` ...................(1)

and `(-4)^4-(-4)^3-13(-4)^2+s(-4)+t=0`

or `256+64-208-4s+t=0`

i.e. `-4s+t=-112` ...................(2)

Subtracting (2) from (1), we get

`-3s+t-(-4s+t)=9-(-112)`

or `-3s+t+4s-t=9+112` or `s=121`

and putting this in (1), we get

`-3xx121+t=9` or `-363+t=9` i.e. `t=372`