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Question
The polynomial px3 + 4x2 – 3x + q is completely divisible by x2 – 1; find the values of p and q. Also, for these values of p and q, factorize the given polynomial completely.
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Solution
Let f(x) = px3 + 4x2 – 3x + q
It is given that f(x) is completely divisible by (x2 – 1) = (x + 1)(x – 1).
Therefore, f(1) = 0 and f(–1) = 0
f(1) = p(1)3 + 4(1)2 – 3(1) + q = 0
p + q + 1 = 0 ...(i)
f(–1) = p(–1)3 + 4(–1)2 – 3(–1) + q = 0
–p + q + 7 = 0 ...(ii)
Adding (i) and (ii), we get,
2q + 8 = 0
q = – 4
Substituting the value of q in (i), we get,
p = –q – 1 = 4 – 1 = 3
∴ f(x) = 3x3 + 4x2 – 3x – 4
Given that f(x) is completely divisible by (x2 – 1)
3x + 4
`x^2 - 1")"overline(3x^3 + 4x^2 - 3x - 4)`
3x3 – 3
– +
4x2 – 4
4x2 – 4
– +
0
∴ 3x3 + 4x2 – 3x – 4 = (x2 – 1)(3x + 4)
= (x – 1)(x + 1)(3x + 4)
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