# Factor p(x) = x3+3x2-16x-48.

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48 factors are 2,3,4,6,8,12,16. So we see for which of these p(x) vanishes.

Obviously, p(-3) = -3^3+3*(-3)^2-16(-3)-48 = -9+9+48-48=0.

p(4) = 4^3+3*4^2-16*x-48 = 64+48-64-48 = 0.

So, (x+3)(x-4) are factors of x^3+3x^2-16x-48.

So, x^3+3x^2-16x-48 = (x+3)(x-4)(x+k). Putting x =0 we get

-48 = 3*-4*k. Or k = -48/(-12 ) = 4.

Therefore (x+3)(x-4)(x+4) are the 3 factors of p(x).

We find the roots of given function as follows:

p(x) = x^3 + 3 x^2 - 16x - 48

= x^2(x + 3) - 16(x + 3)

= (x + 3)(x^2 - 16)

= (x + 3)(x^2 - 4^2)

= (x + 3)(x + 4)(x - 4) ... [Factors of (a^2 - b^2) are (a + b)(a - b)]

Answer:

Factors of p(x) are (x + 3), (x + 4) and (x - 4)