Problem 12 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$y^{2}-8 y+7$$

Step-by-Step Solution

Answer

The trinomial \( y^2 - 8y + 7 \) factors out to \( (y - 1)(y - 7) \)

Step 1: Identify a, b, and c

Firstly, identify the coefficients and the constant in the quadratic equation. The standard form of a quadratic is \( ax^2 + bx + c \) . In this case, 'a' is the coefficient of \( y^2 \), which is 1, 'b' is the coefficient of 'y', which is -8, and 'c' is the constant term, which is 7.

Step 2: Factor

Next, find two numbers that multiply to 'a*c' (which is 7) and add up to 'b' (which is -8). Those numbers are -1 and -7. So, factor the trinomial as: \( (y - 1)(y - 7) \)

Step 3: Checking the factorization using FOIL

Finally, verify that the factoring is correct by multiplying the factors together using the FOIL method. The FOIL method stands for First, Outer, Inner, Last. Multiplying the first terms in each binomial gives \( y^2 \). Multiplying the outer terms gives -7y. Multiplying the inner terms gives -y. Multiplying the last terms gives +7. Adding these together gives \( y^2 - 8y + 7 \), which is the original trinomial, hence confirming the factorization is correct