The expression f(x)=4x^3+ax^2+bx+4, where a and b are constants, has a factor (x-1) and leaves a remainder of -18 when divided by (x-2). Find the...

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If we divide a polynomial `f(x)` by a monomial `(x-d)` with a reminder, `f(x)` becomes expressed as

`f(x)=(x-d)p(x)+c,` where `c` is a reminder (a constant).

Therefore `f(d)=c.` So we know from the conditions that `f(1)=0` and `f(2)=-18.` From this we obtain

`4+a+b+4=0` and `32+4a+2b+4=-18.`

This is the same as `a+b=-8` and `2a+b=-27.` It is a linear system for a and b, and subtracting these equations we obtain `a=-19.` Thus `b=-8-a=11,` and this is the (unique) answer for the problem.

So the complete expression is `f(x) = 4x^3 - 19x^2 + 11x + 4`

Comments

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