Problem 30 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$y^{2}-22 y+72$$

Step-by-Step Solution

Answer

The factorized form of the given trinomial \(y^{2} - 22y + 72\) is \((y - 18)(y - 4)\).

Step 1: Identify the Trinomial

We have been given the trinomial \(y^{2} - 22y + 72\). The next task is to break this down into binomial factors.

Step 2: Factor the Trinomial

To perform the factorization, we need to find two numbers which when multiplied give 72 (the constant term) and when added give -22 (the coefficient with \(y\)). After careful observation, it is seen that those numbers are -18 and -4, since \(-18 \times -4 = 72\) and \(-18 + -4 = -22\). Therefore, the factorized form would be \((y - 18)(y - 4)\).

Step 3: Verify the Factorization using FOIL multiplication

FOIL multiplication stands for First, Outer, Inner, and Last which is a method for multiplying two binomials. Applying this to our factors: \n\nFirst: \(y \times y = y^{2}\)\nOuter: \(y \times -4 = -4y\)\nInner: \(-18 \times y = -18y\)\nLast: \(-18 \times -4 = 72\)\n\nAdding these results together gives, \(y^{2} - 4y - 18y + 72 = y^{2} - 22y + 72\) which is the original trinomial, thereby confirming our factorization.