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.
Isn't 43 a prime number? How can it have factors?
To help Megan understand why her solution does not check, let's go through the process of factoring the expression step by step.
The expression Megan started with is 72x^2 - 43x + 6. She factored it as (9x - 3)(8x - 2).
To check Megan's solution, Jacob used the FOIL principle, which stands for First, Outer, Inner, Last. This method involves multiplying the terms of each binomial in Megan's factored form:
First: Multiply the first terms of each binomial: 9x * 8x = 72x^2
Outer: Multiply the outer terms: 9x * -2 = -18x
Inner: Multiply the inner terms: -3 * 8x = -24x
Last: Multiply the last terms: -3 * -2 = 6
Adding the results together, we have:
72x^2 - 18x - 24x + 6
Combining like terms, the expression simplifies to:
72x^2 - 42x + 6
Jacob's result differs from the original expression, 72x^2 - 43x + 6. Therefore, Megan's solution does not check.
To correctly factor the original expression, let's go through the factoring process again.
We need to find two binomials whose product equals 72x^2 - 43x + 6. One way to do this is by factoring the quadratic expression.
To factor quadratic expressions, we look for two numbers, let's call them a and b, such that their product equals the constant term (6 in this case) and their sum equals the coefficient of the middle term (-43 in this case).
Our goal is to find the values of a and b. In this case, it might be a little tricky since 6 has several possible factor pairs: (1, 6), (-1, -6), (2, 3), and (-2, -3).
After some trial and error, we can determine that the factors that sum to -43 are -42 and -1. Therefore, our values for a and b are -42 and -1.
Now we can rewrite the middle term, -43x, as the sum of -42x and -x, and the original expression becomes:
72x^2 - 42x - x + 6
Grouping the terms, we can factor by grouping:
(72x^2 - 42x) + (-x + 6)
Factoring out the greatest common factor from the first two terms and the last two terms:
6x(12x - 7) - 1(12x - 7)
Notice that we have a common binomial, (12x - 7), in both terms. We can factor it out:
(6x - 1)(12x - 7)
So, the correctly factored expression is (6x - 1)(12x - 7).
We have now factored the original expression correctly.