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Answer to Question #8421 in Algebra for Shreyah mishra

Question #8421
If (x + 3) whole squared is a factor of f(x) = 3 * x cubed + k * x + 6, then find the remainder obtained when f(x) is divided by x - 6
Expert's answer
Let's use the method of undetermined coefficients to find k:

3x³ + kx + 6 = (ax + b)(x + 3)²
3x³ + kx + 6 = (ax + b)(x² + 6x + 9)
3x³ + kx + 6 = ax³ + (6a + b)x² + (9a + 6b)x + 9b

a = 3
0 = 6a + b
k = 9a + 6b
6 = 9b

a = 3
0 = 6*3 + b ==> b = -18
k = 9a + 6b
6 = 9b ==> b = 2/3

Statements b = -18 and b = 2/3 are inconsistent, so (x + 3)² cannot be a factor of 3x³ + kx + 6.

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