What does it mean to factor completely
The Correct Answer and Explanation is:
“Factoring completely” refers to the process of breaking down an expression (typically a polynomial) into a product of simpler factors, where the expression is expressed as a multiplication of irreducible factors. These irreducible factors cannot be factored any further. The goal is to reach the simplest form where no further factoring can be done.
For example, consider the polynomial expression 6×2+11x−356x^2 + 11x – 356×2+11x−35. To factor this completely, you follow these steps:
- Find the GCF (Greatest Common Factor): Look for the largest common factor of all terms. In this case, there is no common factor across all terms, so we proceed to the next step.
- Factor using grouping (if applicable) or apply methods like trial and error or the AC method (splitting the middle term). For 6×2+11x−356x^2 + 11x – 356×2+11x−35, the AC method works:
- Multiply the coefficient of x2x^2×2 (6) by the constant term (-35), which gives -210.
- Find two numbers that multiply to -210 and add up to 11. These numbers are 21 and -10.
- Rewrite the middle term: 6×2+21x−10x−356x^2 + 21x – 10x – 356×2+21x−10x−35.
- Group: (6×2+21x)−(10x+35)(6x^2 + 21x) – (10x + 35)(6×2+21x)−(10x+35).
- Factor each group: 3x(2x+7)−5(2x+7)3x(2x + 7) – 5(2x + 7)3x(2x+7)−5(2x+7).
- Factor out the common binomial factor: (3x−5)(2x+7)(3x – 5)(2x + 7)(3x−5)(2x+7).
- Check for irreducibility: After factoring, ensure that neither 3x−53x – 53x−5 nor 2x+72x + 72x+7 can be factored further. Since they cannot, the factorization is complete.
Thus, factoring completely means fully simplifying the expression by finding all possible factors and leaving them in their simplest irreducible form.
