Using factor theorem, show that (x - 3) is a factor of x^3 - 7x^2… - Vidyadip

Question

Using factor theorem, show that $(x - 3)$ is a factor of $x^3 - 7x^2 + 15x - 9$, Hence, factorise the given expression completely.

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Answer

Let $p(x) = x^3 - 7x^2 + 15x - 9$
For checking that (x - 3) is a factor of p(x), we find : p(3)
$p(3) = (3)^3 - 7(3)^2 + 15(3) - 9$
$= 27 - 63 + 45 - 9$
$= 72 - 72$
$= 0.$
Hence, $(x - 3)$ is a factor of p(x).
By division of $p(x)$ by $x - 3$, we get the quotient
$= x^2 - 4x + 3$
$\therefore x^3 - 7x^2 +15x - 9$
$= (x - 3) (x^2 - 4x + 3)$
$= (x - 3) (x - 3) (x - 1)$
$= (x - 3)^2 (x - 1).$

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