I'm trying to factor a problem and cannot seem to get it to work out ---
12a^3 + 10a^2 - 8a
I first factored out the gcf of 2 to get 2a(6a^2 + 5a - 4)
From there I keep coming up with answers that are incorrect when I check them -- any suggestions please?
2a(3a+4)(2a-1)
12a³ + 10a² - 8a = 2a ( 6a² + 5a - 4 ) =
2a ( 6a² - 3a + 8a - 4 ) = 2a [ ( 6a² - 3a ) + ( 8a - 4 ) ] =
2a [ 3a ( 2a - 1 ) + 4 ( 2a - 1 ) ] =
2a [ ( 2a - 1 ) ( 3a + 4 ) ] = 2a ( 2a - 1 ) ( 3a + 4 )
12 a³ + 10 a² - 8 a = 2a ( 2a - 1 ) ( 3a + 4 )
To factor the expression 12a^3 + 10a^2 - 8a, you correctly factored out the greatest common factor (GCF) of 2a, which gives you 2a(6a^2 + 5a - 4). Now, let's focus on factoring the trinomial expression inside the parentheses: 6a^2 + 5a - 4.
We can try factoring this trinomial using the ac method.
First, multiply the coefficient of the quadratic term (6) by the constant term (-4), which gives you -24.
Next, we need to find two numbers that multiply to -24 and add up to the coefficient of the linear term (5). In this case, the numbers are 8 and -3 because 8 multiplied by -3 is -24, and 8 plus -3 is 5.
Now, rewrite the trinomial by splitting the middle term using these numbers: 6a^2 + 8a - 3a - 4.
Group the terms: (6a^2 + 8a) - (3a + 4).
Take out the greatest common factor from each group: 2a(3a + 4) - 1(3a + 4).
Notice that we have a common binomial factor of (3a + 4).
Factor out (3a + 4): (2a - 1)(3a + 4).
So, the fully factored expression becomes 2a(2a - 1)(3a + 4).
To check the factoring, you can expand the factored form and see if it simplifies back to the original expression.
Hope this helps! Let me know if you have any other questions.