To determine if you have completely factored a polynomial, follow these steps:
- Check for Common Factors: Ensure that there are no common factors left in the polynomial. If there are, factor them out.
- Factor Completely: Continue factoring until all factors are prime (i.e., they cannot be factored further).
- Verify the Degree: The degree of the polynomial should match the sum of the degrees of its factors. For example, if you have a quadratic polynomial, it should factor into two linear factors.
- Multiply the Factors: Multiply the factors back together to ensure you get the original polynomial. If the product matches the original polynomial, you have factored it completely.
- Look for Special Forms: Check if the polynomial fits any special factoring forms like difference of squares, perfect square trinomials, or sum/difference of cubes, and factor accordingly.
By following these steps, you can be confident that you have completely factored the polynomial.