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प्रश्न
Prove that (5x - 4) is a factor of the polynomial f(x) = 5x3 - 4x2 - 5x +4. Hence factorize It completely.
उत्तर
If 5x - 4 is assumed to be factor, then x = `4/5` . Substituting this in problem polynomial, we get:
`"f"(4/5) = 5 xx (4/5) xx (4/5) xx (4/5) - 4 xx (4/5) xx (4/5) - 5 xx (4/5) + 4`
`= 64/25 - 64/25 - 4 + 4`
= 0
Hence (5x - 4) is a factor of the polynomial.
Multiplying (5x-4) by x2, we get 5x3 - 4x2, hence we are left with -5x + 4 (and 1st part of factor as x2).
Multiplying (5x - 4) by -1, we get -5x + 4, hence we are left with 0 (and 2nd part of factor as -7x).
Hence complete factor is (5x - 4) (x2-1).
Further factorizing (x2 - 1), we get :
⇒ (x - 1)(x + 1) = 0
Hence answer is (5x - 4)(x - 1)(x + 1) = 0
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