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

Problem

P(x)=x3+3*x2−4

Solution

1. Identify a possible root using the Rational Root Theorem. Testing x=1

P(1)=(1)3+3*(1)2−4

P(1)=1+3−4=0

Since P(1)=0 (x−1) is a factor.

2. Divide the polynomial by (x−1) using synthetic division or long division:

(x3+3*x2−4)/(x−1)=x2+4*x+4

3. Factor the resulting quadratic expression x2+4*x+4

x2+4*x+4=(x+2)*(x+2)

x2+4*x+4=(x+2)2

4. Combine all factors to write the completely factored form of the polynomial:

P(x)=(x−1)*(x+2)2

Final Answer

x3+3*x2−4=(x−1)*(x+2)2

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