Problem 15 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$5 y^{2}-16 y+3$$

Step-by-Step Solution

Answer

The given trinomial \(5y^{2}-16 y+3\) factors to \((5y - 1)(y - 1)\)

Step 1: Identify the 'a', 'b', and 'c' values

The first step is to identify the coefficients and the constant in the trinomial. In the given trinomial \(5 y^{2}-16 y+3\), 'a' is 5, 'b' is -16 and 'c' is 3.

Step 2: Find the numbers that add to 'b' and multiplies to 'ac'

Now, find the two numbers that add up to 'b' and multiplies to 'ac'. 'b' is -16 and 'ac' is 15 (the product of 5 and 3). The numbers -5 and -3 meet the conditions because \(-5 + -3 = -8\) and \(-5 \times -3 = 15\).

Step 3: Replace the middle term

Next, replace the middle term (-16y) of the equation with these two terms: \(5 y^{2}-5 y-3 y+3\)

Step 4: Factor by grouping

Perform factorization by grouping. This leads to: \(y(5y - 1) - 1(5y - 1)\). Then factor out the common binomial which is \(5y - 1\), hence the factorization of the trinomial is \((5y - 1)(y - 1)\)

Step 5: Verify using FOIL method

Finally, verify the result by multiplying the factors using the FOIL (First, Outer, Inner, Last) method. Multiply the first terms, outer terms, inner terms, and last terms of the binomials and sum them up, this should yield the original trinomial: \(5y^{2}-16 y+3\)