Final answer:
The trinomial a²-17ab+72b² is factored into (a-8b)(a-9b). The factorization is confirmed by using the FOIL method which results in the original trinomial expression.
Step-by-step explanation:
To factor the trinomial a²-17ab+72b², we need to find two numbers that multiply to 72 (the coefficient of the b² term) and add to -17 (the coefficient of the ab term). The numbers that satisfy these conditions are -8 and -9 since (-8)(-9) = 72 and (-8) + (-9) = -17. So the trinomial can be factored as (a-8b)(a-9b).
To check the factorization using FOIL multiplication, we multiply the first terms, the outer terms, the inner terms, and the last terms and then add the results:
- First terms: a * a = a²
- Outer terms: a * (-9b) = -9ab
- Inner terms: (-8b) * a = -8ab
- Last terms: (-8b) * (-9b) = 72b²
Adding these up, we get a² - 9ab - 8ab + 72b² = a² - 17ab + 72b², which matches the original trinomial, thereby confirming our factorization is correct.