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Question
f(x) = x3 − 6x2 + 11x − 6, g(x) = x2 − 3x + 2
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Solution
It is given that f(x) = x3 − 6x2 + 11x − 6, and g(x) = x2 − 3x + 2
We have
g(x) = x2 − 3x + 2
` = x^2 - 2x + x + 2`
` = (x - 2) (x-1)`
\[\Rightarrow \left( x - 2 \right)\]
and (x − 1) are factor of g(x) by the factor theorem.
To prove that (x − 2) and (x − 1) are the factor of f(x).
It is sufficient to show that f(2) and f(1) both are equal to zero.
`f(2) = (2)^3 - 6(2)^3 + 11(2) - 6`
` = 8 - 23 + 22 - 6`
` = 30 - 30`
f(2) = 0
And
`f(1) = (1)^3 - 6(1)^2 + 11(1)- 6`
` = 1-6 + 11 - 6`
` = 12 - 12`
f (1) = 0
Hence, g(x) is the factor of the polynomial f(x).
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