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Question
Using factor theorem, show that (x - 3) is a factor of x3 - 7x2 + 15x - 9, Hence, factorise the given expression completely.
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Solution
Let p(x) = x3 - 7x2 + 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
= x2 - 4x + 3
∴ x3 - 7x2 +15x - 9
= (x - 3) (x2 - 4x + 3)
= (x - 3) (x - 3) (x - 1)
= (x - 3)2 (x - 1).
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