Factoring higher-degree polynomials
Factoring Higher-Degree Polynomials
Factoring Higher-Degree Polynomials is a topic in Polynomial Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.7(D), which requires students to determine the linear factors of a polynomial of degree three and four using algebraic methods.
Factoring rewrites a polynomial as a product using the GCF, difference of squares, grouping, and quadratic-form substitution.
Theory
- GCF first — always remove the greatest common factor.
- Difference of squares: \(a^2-b^2=(a-b)(a+b)\).
- Grouping for four terms.
- Quadratic form: substitute for \(x^2\) in \(x^4+bx^2+c\).
Key patterns:
How to factor
- Remove the GCF.
- Count terms: two \(\to\) difference of squares; three \(\to\) trinomial; four \(\to\) grouping.
- Apply the matching pattern.
- Factor again until prime.
Pull out the GCF \(3x^2\).
| \(6x^3-9x^2\) | \(=\) | \(3x^2(2x-3)\) |
Apply the pattern twice.
| \(x^4-16\) | \(=\) | \((x^2-4)(x^2+4)\) |
| \(=\) | \((x-2)(x+2)(x^2+4)\) |
Group in pairs and factor each.
| \(x^2(x+2)+3(x+2)\) | ||
| \(=\) | \((x+2)(x^2+3)\) |
Treat \(x^2\) as the variable.
| \((x^2-1)(x^2-4)\) | ||
| \(=\) | \((x-1)(x+1)(x-2)(x+2)\) |
Common pitfalls
Frequently asked questions
What should you factor out first?
The greatest common factor (GCF).
How do you factor four terms?
Group them in pairs and factor each pair, then factor out the common binomial.
Does \(a^2+b^2\) factor?
Not over the real numbers; only \(a^2-b^2\) does.
When is a polynomial fully factored?
When every factor is prime and cannot be factored further.