QuestionAugust 27, 2025

Using the Fundamental Theorem of Algebra, find all the roots of the polynomial: x^3-x^2+4x-4=0 Real: square Complex: square square

Using the Fundamental Theorem of Algebra, find all the roots of the polynomial: x^3-x^2+4x-4=0 Real: square Complex: square square
Using the Fundamental Theorem of Algebra, find all the roots of the polynomial: x^3-x^2+4x-4=0 Real: square Complex: square square

Solution
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Answer

Real: 1 ### Complex: 2i, -2i Explanation 1. Identify Possible Rational Roots Use the Rational Root Theorem. Possible rational roots are \pm 1, \pm 2, \pm 4. 2. Test Rational Roots Substitute x = 1: 1^3 - 1^2 + 4(1) - 4 = 0. So, x = 1 is a root. 3. Perform Polynomial Division Divide x^3 - x^2 + 4x - 4 by x - 1 to get x^2 + 4. 4. Solve Quadratic Equation Solve x^2 + 4 = 0. Roots are x = \pm 2i.

Explanation

1. Identify Possible Rational Roots<br /> Use the Rational Root Theorem. Possible rational roots are $\pm 1, \pm 2, \pm 4$.<br /><br />2. Test Rational Roots<br /> Substitute $x = 1$: $1^3 - 1^2 + 4(1) - 4 = 0$. So, $x = 1$ is a root.<br /><br />3. Perform Polynomial Division<br /> Divide $x^3 - x^2 + 4x - 4$ by $x - 1$ to get $x^2 + 4$.<br /><br />4. Solve Quadratic Equation<br /> Solve $x^2 + 4 = 0$. Roots are $x = \pm 2i$.
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