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प्रश्न
Show that (2x + 1) is a factor of 4x3 + 12x2 + 11 x + 3 .Hence factorise 4x3 + 12x2 + 11x + 3.
उत्तर
Let 2x + 1 = 0,
then x =
Substituting the value of x in f(x),
f(x) = 4x3 + 12x2 + 11x + 3
`f (-1/2) = 4(-1/2)^3 + 12(-1/2)^2 + 11(-1/2) + 3`
= `4(-1/8) + 12(1/4) + 11(-1/2) + 3`
= `4(1)/(2) + 3 - (11)/(2) + 3`
= (6) – (6)
= 0
∵ Remainder = 0
∴ 2x + 1 is a factor of
4x3 + 12x2 + 11x + 3
Now dividing f(x) by 2x + 1, we get
`2x + 1")"overline(4x^3 + 12x^2 + 11x + 3)("2x^2 + 5x + 3`
4x3 + 2x2
– –
10x2 + 11x
10x2 + 5x
– –
6x + 3
6x + 3
– –
x
∴ 4x3 + 12x2 + 11x + 3
= (2x + 1)(2x2 + 5x + 3)
= (2x + 1)[2x2 + 2x + 3x + 3]
= (2x + 1)[2x(x + 1) + 3(x + 1]
= (2x + 1)[(x + 1)(2x + 3)]
= (2x + 1)(x + 1)(2x + 3).
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