Special Cases: Perfect Square Trinomials and Prime Polynomials
Perfect square trinomials and prime polynomials are special cases in factoring for Grade 9 Algebra 1 (California Reveal Math, Unit 9: Polynomials). A perfect square trinomial matches a^2 + 2ab + b^2 = (a+b)^2 or a^2 - 2ab + b^2 = (a-b)^2. For example, x^2 + 10x + 25 = (x+5)^2 because sqrt(25) = 5 and 2(1)(5) = 10. A prime polynomial like x^2 + 3x + 7 has no integer factor pair of 7 summing to 3 and cannot be factored.
Key Concepts
A perfect square trinomial fits the pattern: $$a^2 + 2ab + b^2 = (a + b)^2 \quad \text{or} \quad a^2 2ab + b^2 = (a b)^2$$.
A prime polynomial has no factor pairs of $c$ (or $ac$) with integer coefficients whose sum equals $b$. It cannot be factored over the integers.
Common Questions
How do you recognize a perfect square trinomial?
Check: are the first and last terms perfect squares? Does the middle term equal 2 times the product of their square roots? For x^2 + 10x + 25: sqrt(25) = 5, and 2(1)(5) = 10. Yes — it's (x+5)^2.
How do you factor x^2 - 6x + 9?
sqrt(9) = 3, and |middle term| = 6 = 2(1)(3). Negative middle term means (a-b)^2 pattern. So x^2 - 6x + 9 = (x-3)^2.
How do you determine if a trinomial is prime?
List all integer factor pairs of c (the constant). If none have a sum equal to b (the middle coefficient), the polynomial is prime. For x^2 + 3x + 7: factor pairs of 7 are (1,7) sum 8 and (-1,-7) sum -8. Neither sums to 3, so it is prime.
Should you exhaust all factor pairs before calling a polynomial prime?
Yes. Always check every integer factor pair of c (or ac when a is not 1) and verify by expansion. Declaring a polynomial prime prematurely is a common error.
What is the first step before checking for perfect square trinomials?
Always factor out the GCF first. For 2x^2 + 12x + 18: GCF = 2 gives 2(x^2 + 6x + 9), which then factors as 2(x+3)^2. Missing the GCF leads to incomplete factoring.