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Question
Find the value of ‘k’ if (x – 2) is a factor of x3 + 2x2 – kx + 10. Hence determine whether (x + 5) is also a factor.
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Solution
p(x) = x^3 + 2x^2 - kx + 10
For (x - 2) to be the factor of p(x) = x3 + 2x2 – kx + 10
p(2) = 0
Thus (2)3 + 2(2)2 – k(2) + 10 = 0
⇒ 8 + 8 – 2k + 10 = 0
⇒ k = 13
Thus p(x) becomes x3 + 2x2 –13x + 10
Now, (x+5) would be the factor of p(x) iff p(–5) = 0
p(–5) = (–5)3 + 2(–5)2 – 13(–5) + 10 = –125 + 50 + 65 + 10 = 0
Thus, (x + 5) is also a factor of p(x).
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