To factor cubic polynomials, you can follow these general steps:
Steps to Factor Cubic Polynomials
- Find a root (zero) of the cubic polynomial
Use the Rational Root Theorem to list possible rational roots. These are factors of the constant term divided by factors of the leading coefficient. Test these candidates by substitution to find a root aaa, so that (x−a)(x-a)(x−a) is a factor
- Divide the cubic polynomial by the linear factor(x−a)(x-a)(x−a)
Use polynomial long division or synthetic division to divide the cubic polynomial by (x−a)(x-a)(x−a). This gives a quadratic polynomial as the quotient
- Factorize the quadratic polynomial obtained
Factor the quadratic polynomial using methods such as factoring by grouping, splitting the middle term, or using the quadratic formula if necessary. Sometimes the quadratic may be irreducible over the reals
- Express the cubic polynomial as the product of its factors
Combine the linear factor and the quadratic factor to write the full factorization. If the quadratic can be factored further, factor it into linear terms
Additional Tips
- Factoring by grouping can sometimes be used directly on the cubic polynomial by splitting it into two groups and factoring out common terms
- If no rational roots are found, the cubic polynomial may not factor nicely over the rationals; in this case, use the cubic formula or numerical methods to find roots
- Every cubic polynomial with real coefficients can be factored into one linear factor and one quadratic factor
Example
Factorize f(x)=x3−5x2+4x−20f(x)=x^3-5x^2+4x-20f(x)=x3−5x2+4x−20:
- Group terms: (x3−5x2)+(4x−20)(x^3-5x^2)+(4x-20)(x3−5x2)+(4x−20)
- Factor each group: x2(x−5)+4(x−5)x^2(x-5)+4(x-5)x2(x−5)+4(x−5)
- Factor out common binomial: (x−5)(x2+4)(x-5)(x^2+4)(x−5)(x2+4)
Since x2+4x^2+4x2+4 cannot be factored further over the reals, this is the factorization
. This approach provides a systematic method for factoring cubic polynomials by finding roots, dividing, and then factoring the resulting quadratic.
