Factorization is my worst nightmare!?

Question

How do I factor:

4x^2-3x-1=?

3x^2-4x+1=?

I dont even get this is sooo confusing to me, cause factoring is super hard, and dont get why anyone could get this comlicated maths. Please, I dont want stright anwsers please show step by step whys theres a (x-1)?

Answer 1

Don't panic! OK, to factor a quadratic, look first to see what the factors of the constant are. In the first example, the constant is -1, so the factors must be -1 and 1 since
-1*1 is the only way to get -1. Next look at the coefficient of the quadratic term. In this case, it's 4. You can get 4 by multiplying 4 x 1 or 2 x 2. So let's assume that the correct coefficients are 4 and 1. You will have:

(4x ** 1)(x ** -1), where the ** represent either + or -. Now you know that the only way to get -1 (in the original equation) is to multiply -1 and 1, so you have two possibilities:

(4x -1)(x+1) or (4x+1)(x-1).

If you look at these, the first factor gives you 4x^2+3x-1.

This isn't what you wanted; what about the second? It gives 4x^2 -3x -1. Aha! That's the answer! If neither of these had worked you would have tried (2x-1)(2x+1) to see if this would've worked; if not, then the equation couldn't be factored.

For the second one you go through the same process; you know that you have to have (3x**1)(x**1), since that's the only way to get 3 as a coefficient for the quadratic term and 1 x 1 or -1*-1 are the only ways to get 1 for the constant term. If you use a "+" sign in both terms you get +4x; if you use "-" then you get -4x. Aha! You want the "-", so you end up with (3x-1)(x-1)

Hope this helps!

Answer 2

Since the last part of each polynomial is a 1, the factors will be of the form ( ±1)( ±1)

The first polynomial will factor with one having (plus) and one having (minus) to achieve a result of -1.

So put that in the parentheses ( + 1)( -1). Now, the other components that we will place in the factor must both multiply to get 4x^2 and combine to get -3x. Since we need 4x^2, we know that we have only 2 choices for factors (2x,2x) or (4x,x). (2x,2x) will not work because they will cancel and not give us the -3x we need. So we pick (4x,x). Place the 4x in the first factor so that it will multiply by (-1). Place the x in the 2nd factor so it is multiplied by (1) You get (4x+1)(x-1)

The next one...
Again the polynominal dictates what goes in the parentheses.
The last part of each factor again must be (1). The -4x requires that these (ones) are both negative. So we have so far... ( -1) ( -1)
The 3x^2 has one set of factors (3x,x) When we put them in the factors, this is a correct factorization. (3x-1)(x-1)

Happy factoring.

Answer 3

i was taught a "trick" in factoring these kind of problems. i dont know if i wil be able to explain it here well but here goes...

1)multiply the first and last coefficients, in this case, (4)(-1)=-4
2)think of factors of -4 that will give the second coefficient, -3, when you add it. (-4)(1)=-4 and -4+1=-3
3)rewrite the equation using those factors:
--> 4x^2-4x+1x-1
4) i dont know how to explain it but do this:
--> 4x(x-1) + 1(x-1)
5) after factoring and you end up with the same thing inside the parenthesis, you're pretty much done:
--> (4x+1)(x-1)

*if you dont get the same thing inside the parenthesis, check the factors you chose. or maybe you have to change the position of the factors you chose to rewrite the equation in #3.

for the next problem...
3x^2 - 4x +1
3x^2 +3x+1x+1
3x(x+1) + 1(x+1)
factors are: (3x+1)(x+1)

Answer 4

you must know that when an expression has a root called "a" it is divisible by (x-a)

look at the first 4x^2-3x-1
this polynome has two roots x=1 x=-1/4
so you can write 4x^2-3x-1 = (x-1) (x+1/4)*k and seeing that the term x^2 has a coefficient 4
the result is 4x^2-3x-1 = 4(x-1) (x+1/4)

3x^2-4x+1 roots are x=1 x= 1/3
I let you make the end
result is 3(x-1) (x-1/3)

Answer 5

The equations equal zero ; we are solving for x .
Since any number times zero equals zero , any x
that makes a factor zero may be a solution .

( ... ± ... ) ( ... ± ... ) = 0

This works for the first term :

( 4x ± ... ) ( x ± ... ) = 0

Only one choice for the third term : 1 times -1
but the signs must work as well .

( 4x + 1 ) ( x - 1 ) = 0

It is now clear that either -¼ or 1 will make this true .

Check in the original equation :

4 ( -¼ ) ² - 3 ( -¼ ) - 1 = 0

4 ( 1 ) ² - 3 ( 1 ) - 1 = 0

Answer 6

4x^2 - 3x - 1 = (4x + 1)(x - 1)
4x.........1lx
x..........-1l-4x
____________
4x^2....-1l-3x

the two rows of dots
draw a line to join 4x and -1 together and another from x to 1
it will look like a cross
it means cross-multiplying the 1st term and the last term in two columns
the 3 l are supposed to be one straight line cutting through the vertical line
on the right hand side, it is the result from cross-multiplying
add the products of the third column. If the result is the same as the second term of the original expression, then the factors will be those shown in the first two rows.

Answer 7

All of your respondents are fine, but this format is difficult for you to see exactly what is going on. Google some of the algebra sites to see how they work. Probably where you are lost is the term, to "factor" and that means to express as a product, which really means no plus or minus signs OUTSIDE a set of parens.
So the stuff has to look like those answers you are trying to get, i.e,

( )( ) and when you multiply them out, you get those trinomials = to zero

(there is a phantom times dot between those two ( )( )

Find the site that is Math for Morons like us... Google Algebra+factoring, and see what you come up with... nice site.