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You also may use the undetermined coefficients method as alternative method, such that:
- write the reminder theorem using the fact that the reminder is a constant since the divisor is a linear binomial
`4x^5 - 3x^2 - 4x + 1 = (x + 1)(ax^4 + bx^3 + cx^2 + dx + e) + f`
`4x^5 - 3x^2 - 4x + 1 = ax^5 + bx^4 + cx^3 + dx^2 + ex + ax^4 + bx^3 + cx^2 + dx + e + f`
Grouping the terms yields:
`4x^5 - 3x^2 - 4x + 1 = ax^5 + x^4(b + a) + x^3(c + b) + x^2(d + c) + x(e + d) + e + f`
Equating the coefficients of like powers, yields:
`a = 4`
`b + a = 0 => b = -a => b = -4`
`c + b = 0 => c = -b => c = 4`
`d + c = -3 => d + 4 = -3 => d = -7`
`e + d = -4 => e - 7 = -4 => e = 3`
`e + f = 1 => f = 1 - 3 => f = -2`
Hence, since the reminder is f, yields that `f = -2` and you need to validate the answer `a)-2.`
A polynomial `f(x)` when divided by a linear polynomial `x-a` , its remainder is given by `f(a)` .
Here, `4x^5-3x^2-4x+1` is divided by `x+1` . so, its remainder is given by f(-1).
Therefore, the right choice is option a).
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