Problem 80 - Algebra and Trigonometry

Question

Factor completely, or state that the polynomial is prime. $$x^{3}+2 x^{2}-x-2$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(x^{3}+2x^{2}-x-2\) is \((x-1)(x+1)(x+2)\)

Step 1: Identify the Polynomial

The given polynomial is \(x^{3}+2 x^{2}-x-2\). It is a cubic polynomial since the highest power of \(x\) is 3.

Step 2: Look for Possible Factors

Since the polynomial is cubic, it may have up to three factors. In order to find potential factors, check if any whole numbers are roots. They are typically between -10 and 10. For example, 1 is a root as when \(x =1\), the polynomial becomes: \(1^{3}+2(1^{2})-(1)-2 = 0\). It indicates that \(x - 1\) is a factor.

Step 3: Perform Polynomial Division

Next, perform polynomial division using the identified root to simplify the polynomial. When you divide the given polynomial by \((x-1)\), the result is \((x^{2}+3x+2)\).

Step 4: Continue Finding Factors

The quotient \((x^{2}+3x+2)\) is a quadratic polynomial. You can factor it further. Factoring \((x^{2}+3x+2)\) gives \((x+1)(x+2)\). So, \((x+1)\) and \((x+2)\) are also factors.

Step 5: Final Answer

Finally, all the factors of the given polynomial are \((x-1)\), \((x+1)\), and \((x+2)\). The complete factorization of the polynomial \(x^{3}+2x^{2}-x-2\) is \((x-1)(x+1)(x+2)\).