Problem 29 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$y^{2}-18 y+65$$

Step-by-Step Solution

Answer

The factorization of the trinomial \(y^{2}-18 y+65\) is \((y-13)(y-5)\)

Step 1: Determine a, b, and c

From the trinomial \(y^{2}-18 y+65\), determine that a=1, b=-18, and c=65

Step 2: Find two numbers

Find two numbers that multiply to give ac (a*c = 1*65 = 65) and add up to give b (-18). After careful consideration, these numbers are -13 and -5, since -13*-5 = 65 and -13 + -5 = -18.

Step 3: Factor the trinomial

Write the trinomial as \((y-13)(y-5)\). These are the factors of the given trinomial.

Step 4: Check the factorization using FOIL

Use FOIL to check the factors: \((y-13)(y-5)\) gives \(y^{2} - 5y - 13y + 65\), which simplifies back to the original trinomial \(y^{2}-18 y+65\).

Step 5: Conclusion

If the result of the FOIL process is equivalent to the original trinomial, the factorization is correct.