Megan factored the expression 12x^2+13x-14 as (3x+2)(4x-7). But when Jacob applied the foil principle and multiplied out(3x+2)(4x-7), he got 12x^2-13x-14; 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.
12x^2+13x-14=
12x^2-8x+21x-14=(12x^2-8x)+(21x-14)=
4x(3x-2)+7(3x-2)=(3x-2)(4x+7)
To help Megan understand why her factoring doesn't check, let's analyze the process in a step-by-step manner.
1. Megan initially factored the expression 12x^2 + 13x - 14 and came up with (3x + 2)(4x - 7).
2. Jacob applied the FOIL principle to multiply (3x + 2)(4x - 7).
FOIL stands for First, Outer, Inner, Last, and represents the multiplication of all combinations of the terms in each binomial expression.
Here's how Jacob expanded the expression:
First: 3x * 4x = 12x^2
Outer: 3x * -7 = -21x
Inner: 2 * 4x = 8x
Last: 2 * -7 = -14
When we combine these terms, we get 12x^2 - 21x + 8x - 14, which simplifies to 12x^2 - 13x - 14.
From Megan's factored form (3x + 2)(4x - 7) and Jacob's expansion (12x^2 - 13x - 14), we see that Megan's solution is indeed incorrect.
To help Megan understand the mistake, let's go through the correct factoring process:
Start with the expression: 12x^2 + 13x - 14.
We need to find two binomial expressions whose product equals the given expression.
The first term, 12x^2, can only be factored as (3x)(4x) or (6x)(2x).
The last term, -14, can be factored as either (-1)(14) or (1)(-14).
Now, we need to find a combination of the above factors that will give us the middle term, 13x, in the original expression.
Let's try the factors (3x)(4x) and (-1)(14):
(3x + 2) * (4x - 7) =
First: (3x)(4x) = 12x^2
Outer: (3x)(-7) = -21x
Inner: (2)(4x) = 8x
Last: (2)(-7) = -14
When we combine these terms, we get 12x^2 - 21x + 8x - 14, which simplifies to 12x^2 - 13x - 14.
This matches the original expression, so (3x + 2)(4x - 7) is indeed the correct factored form.
Therefore, the mistake Megan made was an error in finding the correct combination of factors for the middle term.