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Question
The polynomial 3x3 + 8x2 – 15x + k has (x – 1) as a factor. Find the value of k. Hence factorize the resulting polynomial completely.
Solution
Given, P(x) = 3x3 + 8x2 – 15x + k
Put x – 1 = 0
x = 1
Now, P(1) = 3(1)3 + 8(1)2 – 15(1) + k = 0
`\implies` 3 + 8 – 15 + k = 0
`\implies` – 4 + k = 0
`\implies` k = 4
Hence, k = 4
Factorization:
P(x) = 3x3 + 8x2 – 15x + 4
`x - 1")"overline(3x^3 + 8x^2 - 15x + 4)(3x^2 + 11x - 4`
3x3 – 3x2
– +
11x2 – 15x
11x2 – 11x
– +
– 4x + 4
– 4x + 4
+ –
∴ 3x3 + 8x2 – 15x + 4 = (x – 1)(3x2 + 11x – 4)
= (x – 1)(3x2 + 12x – x – 4)
= (x – 1)[3x(x + 4) – 1(x + 4)]
= (x – 1)(3x – 1)(x + 4)
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