# Factoring trinomials

• Oct 18th 2012, 07:02 PM
linds12
Factoring trinomials
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:

3x3+4xy+y2

6x2-7xy-5y2

15x2-31xy+10y2

Thank you
• Oct 18th 2012, 07:29 PM
topsquark
Re: Factoring trinomials
Quote:

Originally Posted by linds12
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:

3x3+4xy+y2

6x2-7xy-5y2

15x2-31xy+10y2

Thank you

This sounds awfully like you are giving us problems with your *graded* homework problems. Do NOT do this again or you will be banned.

-Dan

(For the record, the first problem cannot be factored. Typo?)
• Oct 18th 2012, 07:57 PM
querti09
Re: Factoring trinomials
I'll give an explanation for the second problem: $\displaystyle 6x^{2}-7xy-5y^{2}$
1) This is a trinomial of the form $\displaystyle ax^{2}+bx+c$
because we can identify an $\displaystyle x^{2}$
in the first term and an $\displaystyle x$
in the second one and the coefficient isn't the number $\displaystyle 1$
2) We convert this polynomial to the form $\displaystyle x^{2}+bx+c$
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 $\displaystyle x^{2}$, in this case $\displaystyle 6$, in the following way:
a)We will multiply the first term by $\displaystyle 6$, so the first term of the new polynomial is going to be: $\displaystyle 36x^{2}$, because $\displaystyle 6\cdot6x^{2}=36x^{2}$
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: $\displaystyle -7y(6x)$
This can be confusing but it's just reorganizing the letters and the quantities in a way that seems like the polynomial $\displaystyle x^{2}+bx+c$
We put into parenthesis the $\displaystyle (6x)$
because we know that $\displaystyle (6x)^{2}=36x^{2}$, and we are looking for a pattern that is like $\displaystyle x^{2}+bx+c$
So $\displaystyle 36x^{2}$
here is the same as $\displaystyle x^{2}$, the first term in the polynomial that we take as a model: $\displaystyle x^{2}+bx+c$
And $\displaystyle 6x$
here is the same as the $\displaystyle x$ in $\displaystyle bx$, the second term in the polynomial that we take as a model: $\displaystyle x^{2}+bx+c$
c)For the last term, we just multiply by $\displaystyle 6$
, as we have done with in a) and we get $\displaystyle 30y^{2}$
, because $\displaystyle 6\cdot-5y^{2}=-30y^{2}$
Now, we have our new polynomial: $\displaystyle 36x^{2}-7y(6x)-30y^{2}$
We have to solve this as if it were $\displaystyle x^{2}+bx+c$
We create two sets of parenthesis and we plug the square root of the first term in both, we consider $\displaystyle 36x^{2}$
as one quantity squared.
$\displaystyle (6x )(6x )$
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.
$\displaystyle (6x - )(6x+ )$
and now we look into $\displaystyle 36x^{2}-7y(6x)-30y^{2}$
what 2 quantities you substract to get $\displaystyle 7y$ AND you multiply to get $\displaystyle 30y^{2}$
Those quantities are $\displaystyle 10y$ and $\displaystyle 3y$
You plug those quantities, and then you get $\displaystyle (6x-10y)(6x+3y)$
All you have to do now is to divide this factorization by $\displaystyle 6$
, the quantity by which we multiplied everything at the beginning.
we know that $\displaystyle 2\cdot3=6$
so instead of this: $\displaystyle \frac{(6x-10y)(6x+3y)}{6}$
we have this: $\displaystyle \frac{(6x-10y)(6x+3y)}{2\cdot3}$
And we can divide $\displaystyle \frac{(6x-10y)}{2}$
and $\displaystyle \frac{(6x+3y)}{3}$
to get our final answer: $\displaystyle (3x-5y)(2x+y)$