Mathematics
If (x - 2) is a factor of (x3 + 2x2 - kx + 10), find the value of k. Hence, determine whether (x + 5) is also a factor of the given expression.
Factorisation
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Answer
Let, f(x) = x3 + 2x2 - kx + 10.
Since, x - 2 is factor of f(x), thus f(2) = 0.
∴ (2)3 + 2(2)2 - k(2) + 10 = 0
⇒ 8 + 2(4) - 2k + 10 = 0
⇒ 8 + 8 - 2k + 10 = 0
⇒ 26 - 2k = 0
⇒ 2k = 26
⇒ k =
⇒ k = 13.
f(x) = x3 + 2x2 - 13x + 10
⇒ x + 5 = 0
⇒ x = -5.
f(-5) = (-5)3 + 2(-5)2 - 13(-5) + 10
= -125 + 2(25) + 65 + 10
= -125 + 50 + 65 + 10
= 125 - 125
= 0.
Since f(−5) = 0, thus (x + 5) is a factor of f(x).
Hence, value of k = 13 and x - 5 is factor of x3 + 2x2 - 13x + 10.
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