Question

3a^3+33a^2+90a factor the trinomial completely

Answer 1

First step: do the coefficients of the trinomial have any common factors?

Answer 2

? do they?

Answer 3

Look at the 3 numbers: 3, 33, 90. Any common factors there? Is there a number which evenly divides all 3?

Answer 4

3?

Answer 5

Good answer. What do you have left after you divide everything by 3?

Answer 6

1, 11, 30?

Answer 7

@tkhunny

Answer 8

3a^3+33a^2+90a

We're shopping for common factors. You already identfied '3'. This gives

3(a^3+11a^2+30a)

Let's keep looking and find more common factors. See anything?

Answer 9

idk

Answer 10

...?

Answer 11

This is an important skill. You have to get good at it.

'a' is also common to all terms. This gives:

3a(a^2+11a+30)

One more time -- Any more common factors? The same factor in every term inside the parentheses.

Answer 12

no

Answer 13

Perfect. Now we jsut have the nasty business of factoring a trinomial into a pair of binomials - if we can.

3a(a^2+11a+30) = 2a(a + ___)(a + ___)

It there a pair of numbers such that
1) When multipled give 30 and
2) When added give 11

You can see the 11 and 30 in the trinomial.

Answer 14

3 and 8?

Answer 15

5 and 6

Answer 16

sorry

Answer 17

Not quite. You hit the 11 but not the 30. They must be factors of 30.

The only factors of 30 are:

1*30
2*15
3*10
5*6

Add them

1+30 = 31 -- No good.
2 + 15 = 17 -- No good.
3 + 10 = 13 -- No good.
5 + 6 = 11 -- That's it.

We are done.

3a^3+33a^2+90a = 3a(a+5)(a+6)

Answer 18

that first line should be
\[3a(a^2+11a+30 = 3a(a + __)(a+ __)\]just an accidental typo, I'm sure

Answer 19

awesome

Answer 20

sort of like my leaving off the trailing ')' in the left hand side :-)

Answer 21

Good call. 3 should not magically turn into 2.

Answer 22

I mention it only to prevent the attentive from getting confused by the magic transformation...thanks for stepping in, I got called away

Answer 23

let me reinforce that statement that this is something that you need to practice and get proficient or you're going to have a miserable experience...

Answer 24

one way you can practice is by just making up random pairs of things to multiply together, and see what you get. write them down on one side of an index card, and on the other side, write down the multiplied version. make a handful every day, and go through the stack looking at the written out side, trying to identify the factors. there's nothing magical here requiring deep mathematical insight, just some practice.