Factor x^3+4x^2-3x-18

Problem

x3+4*x2−3*x−18

Solution

1. Test for rational roots using the Rational Root Theorem. We test factors of the constant term −18 such as ±1,±2,±3

2. Evaluate the polynomial at x=2

2+4*(2)−3*(2)−18=8+16−6−18=0

3. Identify that since x=2 is a root, (x−2) is a factor.

4. Perform synthetic division to divide x3+4*x2−3*x−18 by (x−2)

(x3+4*x2−3*x−18)/(x−2)=x2+6*x+9

5. Factor the resulting quadratic expression x2+6*x+9 This is a perfect square trinomial.

x2+6*x+9=(x+3)2

6. Combine all factors to write the final factored form.

Final Answer

x3+4*x2−3*x−18=(x−2)*(x+3)2

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