Megan factored the expression3x^2-23x+14 as (x-2)(3x-7) . But when Jacob applied the FOIL principle and multiplied out (x-2)(3x-7) , he got 3x^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.
Look at example 7 to understand the definition of prime.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut28_facttri.htm
To understand why Megan's solution doesn't appear to check, let's go through the steps she followed to factor the expression:
1. Megan factored the expression 3x^2-23x+14 as (x-2)(3x-7).
Now, let's apply the FOIL principle to multiply out (x-2)(3x-7) and see if we get the original expression:
FOIL stands for First, Outer, Inner, Last, and is a method used to multiply two binomials.
1. First term: Multiply the first terms of both binomials -> x * 3x = 3x^2.
2. Outer term: Multiply the outer terms of both binomials -> -2 * 3x = -6x.
3. Inner term: Multiply the inner terms of both binomials -> x * -7 = -7x.
4. Last term: Multiply the last terms of both binomials -> -2 * -7 = 14.
Now, let's combine the terms obtained from the FOIL calculation:
3x^2 - 6x - 7x + 14 = 3x^2 - 13x + 14.
As we can see, the result we obtained here is different from the original expression. This means that Megan's solution does not check, and the factorization (x-2)(3x-7) is incorrect.
To correctly factor the original expression, we can use a different approach. We need to find two numbers whose product is the product of the coefficient of x^2 (which is 3) and the constant term (which is 14), and whose sum is the coefficient of x (which is -23).
The two numbers that satisfy these conditions are -2 and -7. So, we can factor the expression as follows:
3x^2 - 23x + 14 = 3x^2 - 2x - 21x + 14.
Now, we can group the terms:
(3x^2 - 2x) - (21x - 14).
From the first group, we can factor out a common factor of x:
x(3x - 2).
From the second group, we can factor out a common factor of -7:
-7(3x - 2).
Combining the two terms, we get:
x(3x - 2) - 7(3x - 2).
Now, we can see that (3x - 2) is a common factor, so we can factor it out:
(3x - 2)(x - 7).
Therefore, the correct factorization of the original expression 3x^2 - 23x + 14 is (3x - 2)(x - 7).
Note: The expression is not prime, as it can be factored.