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Show that \( (x-1) \) is a factor of \( \left(x^{10}-1\right) \) and also of \( \left(x^{11}-1\right) \).
Given:
Given expressions are $x^{10}-1$ and $x^{11}-1$.
To do :
We have to show that $x-1$ is a factor of the given expressions.
Solution :
Factor Theorem:
The factor theorem states that if $p(x)$ is a polynomial of degree $n >$ or equal to 1 and $‘a’$ is any real number, then $x-a$ is a factor of $p(x)$, if $p(a)=0$.
Let $P(x)=x^{10}-1$ and $Q(x)=x^{11}-1$.
We have to equate $x-1 = 0$
$x = 1$
Therefore,
$P(1) =1^{10}-1$
$=1-1$
$=0$
$Q(1) =1^{11}-1$
$=1-1$
$=0$
Therefore, $x-1$ is a factor of $x^{10}-1$ and also $x^{11}-1$.
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