A basic question from factorisation in which we have been asked to show that (x−3) is a factor of x3−7x2+15x−9. Hence we are to factorize the polynomial completely
ML Aggarwal, Avichal Publication, Factorisation, chapter 6, question no 12
Rajan@2021Guru
Show that (x−3) is a factor of x 3 −7x 2 +15x−9. Hence factorize x 3 −7x 2 +15x−9.
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Let us assume
x−3=0⇒x=3
Now, f(3)=(3)
3
−7(3)
2
+15(3)−9
=27−63+45−9=72−72=0
∴ By remainder theorem,
(x−3) is a factor of f(x)
Now, dividing f(x) by (x−3), we get
x
3
−7x
2
+15−9=(x−3)(x
2
−4x+3)
=(x−3)(x
2
−x−3x+3)
=(x−3){x(x−1)−3(x−1)}
=(x−3)(x−3)(x−1)
=(x−3)
2
(x−1)