What number should be subtracted from $x^{2} + x + 1$ so that the resulting polynomial is exactly divisible by $(x - 2)$ .

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Solution

Step: Find the number

Let's assume the required number to be k.
$∴ f(x)= x^{2} +x+1-k$

A polynomial f(x) has a factor $(x-a)$ if and only if $f(a) = 0$ .

$f(x)$ has a factor $f(x-2)$ .
$∴ f(2) = 0$
$(2)^{2} + (2) + 1 - k = 0$
$4 + 3 - k = 0$
$k = 7$

Hence, $7$ should be subtracted from $x^{2} +x+1$ .

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What number should be subtracted from x2+x+1 so that the resulting polynomial is exactly divisible by (x - 2).​

Step: Find the number Let's assume the required number to be k. ∴f(x)=x2+x+1-k A polynomial f(x) has a factor (x-a) if and only if f(a)=0. f(x) has a factor f(x-2). ∴f(2)=0 (2)2+(2)+1−k=0 4+3−k=0 k=7 Hence, 7 should be subtracted from x2+x+1.

What number should be subtracted from $x^{2} + x + 1$ so that the resulting polynomial is exactly divisible by $(x - 2)$ .