Polynomial equations (factoring)?

Question

How would you completely factor this?
12x^3 - 60x^2 + 75x = 0

All I can get is 3x(4x^2 - 20x + 25) = 0

Did i go wrong somewhere? My math teacher wants it to be so its something like x(xx+xx)-(xx-x). The x's and +/- signs are just there to show what i mean, don't take it literally plz lol. If you could help and explain the steps i would really appreciate it. thanks in advance

Answer 1

12x^3 - 60x^2 + 75x = 0
= 12x(x^2 - 5x + [25/4])
= 12x(x - [5/2])(x - [5/2])
= 12x(x - [5/2])^2

Answer 2

You have taken out the common factor so far. Now, you need to factor the trinomial inside the second parentheses.

This will be two factors of 4 , which when multiplied by factors of 5 will give you the -20 middle term. Only experience will allow you to see these quickly.

But you can figure them out by listing all factors of 4 in one column and all factors of 5 in another column.

4 * 1 ::: 25 * 1
2 * 2 ::: 5 * 5
1 * 4 ::: 1 * 25
don't ::: -5 * -5
need ::: -1 * -25
- x 's ::: -25 * -1

Multiply any number in 1st colum by any number in 2nd column, then the others in each pair, and add. If this is the middle term, you have the numbers you need. If not, try another group.

If there are any factors that work, they can be found this way.

I'll try a few, then show you which pairs actually work.

4 * 25 (first in each column) + 1 * 1 (second in each) is way too big (101), so I wouldn't use any 25's

How about 4 * 5 + 1 * 5 ? 25 is still too big.

I think I need to use the 2 * 2 in the first column.

Hey, the 2 * 5 + 2 * 5 looks good -- it does add to 20 --but the sign is wrong!

But if I use the 2 * -5 + 2 * -5, it all works, since -10 + -10 = -20, and -5 * -5 = +25.

So that elusive factorization of 4x^2 - 20x + 25 is (2x -5)(2x - 5)
And the whole factored is 3x(2x -5)(2x - 5)

As I said before, experience will let you see these factors much more quickly, oftentimes without even listing the factors in columns. And a few lessons later, you will recognize this as a "perfect square trinomial", and can find the factors for it in seconds flat.

Answer 3

3x(2x-5)(2x-5)

Start by looking at the factors of 4 and 25. You need to come up with ones that when multiplied and added will equal 20. The factors of 4 are (1*4) or (2*2) the factors of 25 are (1*25) and (5 * 5) Since 25 is positive and -20 is negative, you need a form like (ax-b).

so try (x-5)(4x-5) this gives 4x^2 - 25x + 25 no good
so try (2x-1)(2x-25) this gives -52x no good
so try (2x-5)(2x-5) which gives -20x which is what you want.

the trick is to look at the signs and possible factors and just try alternatives until you find what you need.

Answer 4

in 3x(4x^2 - 20x + 25) = 0

4x^2 - 20x + 25 is still factorable

thus, the complete factor is

3x (2x-5)(2x-5) = 0

Answer 5

as without delay as you recognize the way it really is amazingly uncomplicated so i'm really going to do #a million and #2. THE TRICK: get each and each and every and all the stuff interior the parentheses and set them equivalent to 0. Then treatment for x. #a million: (x+a million)(x+10)=0 set the stuff interior the first parenthesis equivalent to 0: x+a million=0 x=-a million set the stuff interior the second one parenthesis equivalent to 0: x+10=0 x=-10 #2 (x+9)^2 set the stuff interior the parenthesis equivalent to 0: x+9=0 x=-9

Answer 6

12x^3-60x^2+75x=0

3x(4x^2-20x+25)=0
3x(2x-5)(2x-5)=0
3x=0
2x-5=0 2x=5
x=0 or x=5/2

Answer 7

answer: 3x(2x+5)(2x-5)

you have to continue with the factoring using trinomial squared.

Answer 8

3x(2x -5)(2x - 5)