SOLUTION: Have tried many ways to factor this: 25x^3y-78x^2y^2+9xy^3= I hope one of you helps me with this. Thanks.

Question 76395: Have tried many ways to factor this: 25x^3y-78x^2y^2+9xy^3= I hope one of you helps me with this. Thanks.
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!

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Note that every term contains an x, so you can factor it out to get:
.

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Notice also that ever term in the parentheses contains a y so y can also be factored
out. When you do that factoring the expression becomes:
.

.
Factoring the polynomial in the parentheses is a little tricky depending on where you are
in your math lessons. If you have not yet studied the quadratic formula, you can use trial
and error along with the following identity:
.
(ax + by)(cx + dy) = (ac)x^2 + (ad+bc)xy + (bd)y^2
.
By comparing the right side of this identity with the polynomial in the parentheses,
you can see that a*c must be 25. This presents several possibilities:
.
a = 25 and c = 1; or a = 1 and c = 25; or a = 5 and c = 5.
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And also by comparing, you can also see that b*d must equal 9. So possible combinations
are:
.
b = 9 and c = 1; or b = 1 and c = 9; or b = 3 and c = 3
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and you must take into consideration whether the signs on these terms are + or -.
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Now you must experiment with the possibilities to find the combination of a, b, c, and d
that will result in (a*d + b*c) equaling -78.
.
I played around and found that a = 1, c = 25, b = -3, and d = -3 will work because
a*d = -3 and b*c = -75 and when combined these two produce -78.
.
Now you can substitute these values for a,b,c, and d into the factored form of (ax + by)(cx + dy).
When you do you get the two factors to be (x - 3y)(25x - 3y). Substituting these factors
into where we left off we get the complete factored form to be:
.

.
You can multiply this out, and if you do, you will find that it checks because the product
is the polynomial that you were originally given to factor.
.
If you are familiar with the quadratic formula you could also apply it to (25x^2 - 78x + 9 = 0)
and find that the solutions are: x = 3 and x = 3/25.
.
This indicates that (x - 3) and (25x - 3) are factors. Don't forget that the y also comes
into play. That means (x - 3y) and (25x - 3y) are the likely factors. This was the clue
I used to guess which combinations of values for a, b, c, and d, were likely to work.
This method certainly reduced the amount of work ...
.
Hope this helps you to understand what might be involved in factoring the given polynomial.
And now that you have the answer, you may be able to find another way of approaching
this problem.
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