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प्रश्न
Show that (x + 4) , (x − 3) and (x − 7) are factors of x3 − 6x2 − 19x + 84
उत्तर
Let f(x) = x3 − 6x2 − 19x + 84 be the given polynomial.
By the factor theorem,
(x+ 4),(x-3)and (x-7) are the factor of f(x).
If f( - 4),f(3)and f(7) are all equal to zero.
Therefore,
`f(-4) = (-4)^3 -6(-4)^2 --9(-4) + 84`
`= -64 - 96 + 76 + 84`
` = -160 + 160`
` = 0`
Also
`f(3) = (3)^3 - 6(3)^2 - 19(3) + 84`
` = 27 - 54 - 57 + 84`
` = 111 - 111`
` = 0`
And
`f(7) = (7)^3 - 6(7)^2 - 19(7)+ 84`
`243 - 294 - 133 + 84`
` = 427 - 427`
` =0 `
Hence, (x + 4),( x - 3)and (x - 7)are the factor of the polynomial f(x).
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