Megan factored the expression (-12x^2+52x-35) as (-2x+5)(6x-7) but when Jacob applied FOIL principle and multiplied out (2x+5)(6x-7) he got (-12x^2+44x-35),thus megan's solution does not appear to check please help megan to understand better.Explain your reasoning and correctly factor the original expression if possible If the expression is a prime please state so
First of all I changed it into
- (12x^2 - 52x + 35)
After 4 tries I had
- (2x-7)(6x-5) or (7-2x)(6x-5) or (2x-7)(5-6x)
(I use a method where I list the factors of the first and last numbers, and then take sum or differences of cross-products
e.g.
3 2 1 ....... 5 7
4 6 12 ...... 7 5
2x5 + 6x7 = 52, my middle term
This is the fastest way to factor trinomials, too bad nobody seems to teach this method any more.
I came up with -(6x-5)(2x-7)
yes,
that was my first answer I gave you.
surely you realize that
-(6x-5)(2x-7) = - (2x-7)(6x-5)
To understand Megan's mistake, let's review the process of factoring and multiplying.
Factoring: In factoring, we aim to break down an expression into its factors. Megan factored the expression (-12x^2+52x-35) as (-2x+5)(6x-7). To check if these factors are indeed correct, we need to multiply them back together and see if we obtain the original expression.
Multiplying: Jacob applied the FOIL method to the expression (2x+5)(6x-7) and got (-12x^2+44x-35), which is different from the original expression Megan started with.
Reasoning: Let's compare the expressions Megan factored and Jacob multiplied to understand Megan's mistake.
Original expression: (-12x^2+52x-35)
Megan's factored expression: (-2x+5)(6x-7)
Jacob's multiplied expression: (-12x^2+44x-35)
By comparing the original expression with Jacob's multiplied expression, we notice that Megan made an error in the coefficient of the x term. Megan factored (-12x^2+52x-35) as (-2x+5)(6x-7), where the coefficient of x is 52, but Jacob's multiplication resulted in 44x.
Correcting Megan's Mistake: To correctly factor the expression (-12x^2+52x-35), we need to find a different pair of factors that, when multiplied, give us the original expression.
To factor trinomials like this one, we look for two numbers whose product is the product of the coefficient of the x^2 term and the constant term. In this case, that is (-12 x -35) = 420.
Possible factor pairs of 420 are:
1 and 420,
2 and 210,
3 and 140,
4 and 105,
5 and 84,
6 and 70,
7 and 60,
8 and 52.
We are looking for a pair of factors whose sum is equal to the coefficient of the x term, which is 52. The pair that satisfies this criterion is 4 and 105 (4 + 105 = 109).
We can now rewrite the expression (-12x^2+52x-35) in terms of these factors:
(-12x^2 + 4x + 105x - 35)
Now, we group the terms:
(4x - 12x^2) + (105x - 35)
We can factor out common terms:
4x(1 - 3x) + 35(3x - 1)
Finally, we can factor out the common factors:
4x(1 - 3x) - 35(1 - 3x)
Hence, the correct factored form of the expression (-12x^2+52x-35) is:
(4x - 35)(1 - 3x)
Note: The factored expression (4x - 35)(1 - 3x) cannot be simplified further, as both factors are prime.