Q. 153.7( 10 Votes )

Without actual division, show that is exactly divisible by .

Answer :

Let f(x) = x3 – 3x2 – 13x + 15


Now, we have


x2 + 2x – 3 = 0


x2 + 3x – x – 3 = 0


(x + 3)(x – 1)


Hence, f(x) will be exactly divisible by x2 + 2x – 3 = (x + 3)(x – 1)


Now,


Using factor theorem,


If (x + 3) and (x – 1) are both the factors of f(x), then f(-3) = 0 and f(1) = 0


Now,


f(-3) = (-3)3 – 3(-3)2 – 13(-3) + 15


= - 27 – 27 + 39 + 15


= - 54 + 54


= 0


And,


f(1) = (1)3 – 3(1)2 – 13(1) + 15


= 1 – 3 - 13 + 15


= 16 - 16


= 0


Now,


Since, f(-3) = 0 and f(1) = 0


Therefore, x2 + 2x -3 divides f(x) completely.


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