Factor 4x^3+4x^2-7x+2

Problem

4*x3+4*x2−7*x+2

Solution

1. Identify potential rational roots using the Rational Root Theorem, which suggests testing factors of the constant term 2 divided by factors of the leading coefficient 4

2. Test the value x=−2 using synthetic division or direct substitution: 4*(−2)3+4*(−2)2−7*(−2)+2=−32+16+14+2=0

3. Divide the polynomial by the factor (x+2) using synthetic division to find the remaining quadratic factor.

4. Result of the division gives the quotient 4*x2−4*x+1

5. Factor the quadratic expression 4*x2−4*x+1 which is a perfect square trinomial (2*x−1)2

6. Combine all factors into the final factored form.

Final Answer

4*x3+4*x2−7*x+2=(x+2)*(2*x−1)2

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