Megan factored the expression 72x^2-43x+6 as (9x-3)(8x-2). But when Jacob applied the FOIL principle and multiplied out (9x-3)(8x-2), he got72x^2-42x+6; thus, Megan’s solution does not appear to check. Why is that? Please help Megan to understand this better. Explain your reasoning and correctly factor the original expression, if possible. If the expression is prime, so state
The expression is prime and cannot be factored.
To understand why Megan's factoring solution does not check, let's analyze the FOIL principle and the steps involved in factoring expressions.
The FOIL principle is a method used to multiply two binomials. It stands for First, Outer, Inner, Last and represents the order in which you multiply the terms. When applying the FOIL method to multiply (9x-3)(8x-2), you follow these steps:
First: Multiply the first terms of each binomial: 9x * 8x = 72x^2
Outer: Multiply the outer terms of each binomial: 9x * -2 = -18x
Inner: Multiply the inner terms of each binomial: -3 * 8x = -24x
Last: Multiply the last terms of each binomial: -3 * -2 = 6
Then, combine the resulting terms:
72x^2 - 18x - 24x + 6
Simplifying this expression gives:
72x^2 - 42x + 6
Thus, Jacob's solution is correct. The discrepancy between Megan's solution and Jacob's solution lies in the term -43x that Megan factored, which should have been -42x.
To correctly factor the original expression 72x^2 - 43x + 6, we need to consider two numbers that multiply to 72 and add up to -43. Since there is no pair of numbers that satisfy these criteria, the expression cannot be factored further.
Therefore, the correct factorization of the original expression is:
72x^2 - 43x + 6 (prime expression)