Algebra I to AP Precalculus · Module 7 of 14
Polynomials & Factoring
Operating on polynomials and the factoring patterns that undo multiplication.
Adding, Subtracting & Multiplying Polynomials
- Add or subtract polynomials by combining like terms — distribute the negative sign across every term when subtracting.
- Multiply binomials with FOIL (First, Outer, Inner, Last) or the distributive property; multiply larger polynomials term by term.
Multiply (x + 3)(x − 5).
Solution. FOIL: x·x + x·(−5) + 3·x + 3·(−5) = x² − 5x + 3x − 15 = x² − 2x − 15.
Factoring GCF & by Grouping
- Always factor out the Greatest Common Factor (GCF) of all terms first.
- Factoring by grouping: split a four-term polynomial into two pairs, factor the GCF from each pair, then factor out the shared binomial.
Factor 6x³ + 9x² by GCF.
Solution. The GCF of 6x³ and 9x² is 3x². Factor it out: 3x²(2x + 3).
Factoring Trinomials
- For x² + bx + c (leading coefficient 1): find two numbers that multiply to c and add to b.
- For ax² + bx + c (a ≠ 1): find two numbers that multiply to a·c and add to b, then factor by grouping.
Factor x² + 7x + 12.
Solution. Find two numbers that multiply to 12 and add to 7: 3 and 4. Factor: (x + 3)(x + 4).
Special Factoring Patterns
- Difference of squares: a² − b² = (a + b)(a − b).
- Perfect square trinomial: a² + 2ab + b² = (a + b)²; a² − 2ab + b² = (a − b)².
- Check for these patterns before trial-and-error factoring — they save time.
Factor x² − 49.
Solution. Recognize the difference of squares (x² and 7²): (x + 7)(x − 7).
Common mistake: A sum of squares like x² + 49 does NOT factor over the real numbers — only a difference of squares factors this way.
Check yourself
2 of this module's 5 quiz questions. Pick your answers, then check them in the app, with an explanation for every question.
1. Multiply (x − 4)(x + 6).
2. Factor the GCF: 8x³ − 12x².