# Without actual division, show that is exactly divisible by .

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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