Algebra 2
Polynomial functions
Factoring higher-degree polynomials
20 practice questions
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Theory + worked examples
Factoring Higher-Degree Polynomials
Common Core Algebra 2 • Standard A-SSE.2 • Polynomial Functions
Factoring Higher-Degree Polynomials is a topic in Polynomial Functions in the Common Core State Standards. It is aligned to Standard A-SSE.2, which requires students to use the structure of an expression to identify ways to rewrite and factor it.
Factoring rewrites a polynomial as a product using the GCF, difference of squares, grouping, and quadratic-form substitution.
Theory
Factoring rewrites a polynomial as a product. Work through a toolkit in order:
- 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 \(x^4+bx^2+c\).
Keep factoring until every factor is prime (fully factored).
Try the tools in order, GCF first.
Factoring by grouping.
Key patterns:
\[a^2-b^2=(a-b)(a+b),\qquad x^2+(p+q)x+pq=(x+p)(x+q)\]
Check by expanding — the product should return the original.
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.
Example 1 — Greatest common factor
Factor \(6x^3-9x^2\).
Solution
Pull out the GCF \(3x^2\).
| \(6x^3-9x^2\) | \(=\) | \(3x^2(2x-3)\) |
Example 2 — Difference of squares
Factor \(x^4-16\).
Solution
Apply the pattern twice.
| \(x^4-16\) | \(=\) | \((x^2-4)(x^2+4)\) |
| \(=\) | \((x-2)(x+2)(x^2+4)\) |
Example 3 — Factor by grouping
Factor \(x^3+2x^2+3x+6\).
Solution
Group in pairs and factor each.
| \(x^2(x+2)+3(x+2)\) | ||
| \(=\) | \((x+2)(x^2+3)\) |
Example 4 — Trinomial in quadratic form
Factor \(x^4-5x^2+4\).
Solution
Treat \(x^2\) as the variable.
| \((x^2-1)(x^2-4)\) | ||
| \(=\) | \((x-1)(x+1)(x-2)(x+2)\) |
Common pitfalls
Always take the GCF first — it simplifies everything after.
Difference of squares can repeat: \(x^4-16\) factors twice.
\(a^2+b^2\) does not factor over the reals.
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.
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Sum and difference of cubes; factoring by grouping
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