Hello, I have a test tomorrow and 150 math problems due tomorrow morning.
I had 4 people explain this to me in 4 different ways, but I always get stuck when I work on problems by myself.
Can someone please explain to me step by step how to factor:
3x^{3}+4xy+y^{2 }6x^{2}-7xy-5y^{2 }15x^{2}-31xy+10y^{2}
^{ Thank you }
I'll give an explanation for the second problem:
1) This is a trinomial of the form
because we can identify an
in the first term and an
in the second one and the coefficient isn't the number
2) We convert this polynomial to the form
because we then can solve (we know the procedure for polynomials of this another form)
We do this by multiplying the whole polynomial by the coefficient that has the , in this case , in the following way:
a)We will multiply the first term by , so the first term of the new polynomial is going to be: , because
b)For the second term, we will write the multiplication but we won't actually do it. We only show the multiplication: So we get:
This can be confusing but it's just reorganizing the letters and the quantities in a way that seems like the polynomial
We put into parenthesis the
because we know that , and we are looking for a pattern that is like
So
here is the same as , the first term in the polynomial that we take as a model:
And
here is the same as the in , the second term in the polynomial that we take as a model:
c)For the last term, we just multiply by
, as we have done with in a) and we get
, because
Now, we have our new polynomial:
We have to solve this as if it were
We create two sets of parenthesis and we plug the square root of the first term in both, we consider
as one quantity squared.
The sign for these factors are as follow: The second one is the sign of the second term in the polynomial that you are factoring now, in this case minus.
The sign of the second factor is the result of multiplying the sign of the second term by the sign of the last term. This rule works always the same. For this case it's positive, because negative times negative is positive.
and now we look into
what 2 quantities you substract to get AND you multiply to get
Those quantities are and
You plug those quantities, and then you get
All you have to do now is to divide this factorization by
, the quantity by which we multiplied everything at the beginning.
we know that
so instead of this:
we have this:
And we can divide
and
to get our final answer: