General Factoring Strategy for Binomials and Trinomials
Grade 9 Algebra 1 students in California Reveal Math (Unit 9: Polynomials) learn a systematic factoring strategy for binomials and trinomials. Always begin by factoring out the Greatest Common Factor. Then identify the type: binomials may factor as a difference of squares (a^2 - b^2 = (a+b)(a-b)), while a sum of squares is not factorable. Trinomials are checked for perfect square patterns first, then factored as (x+p)(x+q) by finding integers with the correct product and sum. Always verify each factor is fully factored.
Key Concepts
When factoring a polynomial completely, follow this decision process:.
Step 1: Always factor out the Greatest Common Factor (GCF) first.
Common Questions
What is the first step in any factoring problem?
Always factor out the Greatest Common Factor (GCF) first. For example, 3x^2 - 75 factors to 3(x^2 - 25) before applying difference of squares to get 3(x+5)(x-5).
How do you factor a binomial difference of squares?
Use a^2 - b^2 = (a+b)(a-b). For 3x^2 - 75: pull GCF of 3 to get 3(x^2 - 25), then factor as 3(x+5)(x-5).
Can a sum of squares be factored?
No. A binomial of the form a^2 + b^2 is not factorable over the integers. Only a difference of squares a^2 - b^2 factors.
How do you recognize and factor a perfect square trinomial?
Check if the first and last terms are perfect squares and the middle term equals 2 times their product. For 2x^2 + 12x + 18: GCF gives 2(x^2 + 6x + 9) = 2(x+3)^2.
How do you factor a general trinomial like x^2 - 5x + 6?
Find two integers p and q where pq = 6 and p + q = -5. The pair (-2, -3) works: pq = 6, sum = -5. So x^2 - 5x + 6 = (x-2)(x-3).
Why must you always check if factors can be factored further?
Some results of the first factoring step are themselves special products. Complete factoring means every factor is prime over the integers with no further factoring possible.