To factor polynomials, you can follow these general methods depending on the polynomial type:
1. Factoring Out the Greatest Common Factor (GCF)
- Identify the greatest common factor of all terms in the polynomial.
- Factor out the GCF from the polynomial.
- Example: 6x3+9x2=3x2(2x+3)6x^3+9x^2=3x^2(2x+3)6x3+9x2=3x2(2x+3)
2. Factoring Trinomials (Quadratic Polynomials)
For a trinomial in the form ax2+bx+cax^2+bx+cax2+bx+c:
- Step 1: Identify coefficients aaa, bbb, and ccc.
- Step 2: Find two numbers that multiply to a×ca\times ca×c and add to bbb.
- Step 3: Rewrite the middle term using these two numbers.
- Step 4: Factor by grouping.
- Example: Factor x2+6x+8x^2+6x+8x2+6x+8:
- Numbers that multiply to 8 and add to 6 are 2 and 4.
- So factors are (x+2)(x+4)(x+2)(x+4)(x+2)(x+4)
3. Factoring by Grouping (For polynomials with 4 terms or after splitting
the middle term)
- Group terms in pairs.
- Factor out the GCF from each group.
- Factor out the common binomial factor.
- Example: Factor 2x2−4x+3x−62x^2-4x+3x-62x2−4x+3x−6:
- Group as (2x2−4x)+(3x−6)(2x^2-4x)+(3x-6)(2x2−4x)+(3x−6)
- Factor each: 2x(x−2)+3(x−2)2x(x-2)+3(x-2)2x(x−2)+3(x−2)
- Factor common binomial: (x−2)(2x+3)(x-2)(2x+3)(x−2)(2x+3)
4. Advanced Techniques
- For higher degree polynomials, try factoring out GCF first.
- Use synthetic division or polynomial division to test possible roots.
- Rearrange terms if needed to create factorable groups.
- Example: Factoring 3y3+18y2+y+63y^3+18y^2+y+63y3+18y2+y+6 by grouping after rearranging terms
Summary of Steps:
- Look for GCF first.
- For trinomials, find two numbers that multiply to a×ca\times ca×c and add to bbb.
- Use factoring by grouping when applicable.
- Verify by expanding factors to check correctness.
These methods cover most polynomial factoring problems from simple quadratics to more complex polynomials
