How to take common factors of powers out?

To factor out a common factor, do this:

1-find the biggest common monomial factor of each term or expression

2-divide the original polynomial by this factor to get the second factor.
The second factor will also be a polynomial.

I will use an easy sample and then target one of your questions.

SAMPLE A:

Factor 5x^2 + 4x

What is the biggest common monomial factor for this polynomial?

In other words, what letter or number or term (it could be a term) is found in BOTH terms or expressions?

How about x alone?

The letter x is found in both terms or expressions, right?

So, we take out x and divide each term by the letter x.

5x^2 divided by x = 5x

4x divided by x = 4

We now put it together and get x * (5x + 4) as the final answer.

This is the same idea that you must use to answer your questions.

I will target question 2.

FACTOR: (3+2x)^3 + (3+2x)^4

We can factor out (3 + 2x)^3 because it can be divided into BOTH expressions.

We now divide each expression by (3 + 2x)^3 just like I did in SAMPLE A above.

(3 + 2x)^3 divided by itself = 1. We know this, right?

(3 + 2x)^4 divided by (3 + 2x)^3

Since they both have the same base quantity (3 + 2x), we can write the entire subtration process this way:

(3 + 2x)^(4 - 3) = (3 + 2x)^1 and anything raised to the first power equals itself.

So, we get (3 + 2x) after subtraction.

Final answer:

(3 + 2x)^3 * (3 + 2x)

(1+3x)^5 + (1+3x)^4

This expression is composed of two terms: (1+3x)^5 and (1+3x)^4.
What is the largest term common to both of these terms? Notice that (1+3x) divides into both, so it's a common factor. But, (1+3x)^2 also divides into both (and is larger than (1+3x)). So keeping going, what is the largest term? (1+3x)^4 divides into both, and it's the largest such term.

So we factor out (1+3x)^4 from them.

(1+3x)^4 * ( something )

To find this 'something', what do we times (1+3x)^4 by to get (1+3x)^5? It should be obvious that it's (1+3x). And what do we times (1+3x)^4 by to get (1+3x)^4? Obviously, 1.

So something = (1+3x) + 1 = 2+3x

Therefore, the answer:

(1+3x)^4 * (2+3x)


Now try the same thing for the second problem yourself. You should get:

(3+2x)^3 as the largest common factor, and therefore:

(3+2x)^3 * (4 + 2x)

You just have to look for factors that the two terms have in common, then divide by the factors and simplify what's left (if possible) So for the first one, (1 + 3x)^4 is a factor of both terms. So you get (1 + 3x)^4 X (1 + 3x + 1) = (2 + 3x) (1 + 3x)^4 Treating the second one similarly you get (4 + 2x)(3 + 2x)^3 If you practice this a bit you'll find it quite easy !

1)
(1 + 3x)^5 + (1 + 3x)^4
= (1 + 3x)^4[(1 + 3x) + 1]
= (1 + 3x)^4(1 + 3x + 1)
= (3x + 1)^4(3x + 2)

2)
(3 + 2x)^3 + (3 + 2x)^4
= (3 + 2x)^3[1 + (3 + 2x)]
= (2x + 3)^3(1 + 3 + 2x)
= (2x + 3)^3(2x + 4)

(1+3x)^5 + (1+3x)^4
The term (1 + 3x) appears 5 times in the first term and appears 4 times in the second term.
So we can take out 4 lots of (1+3x) from each term, that is the common factor. This leaves just 1 lot of (1+3x) in the first term and no (1+3x)'s in the second term.
Mathermetically it is written:-
(1+3x)^4*{(1+3x)^1 + (1+3x)^0}

Remembering that any number with an index of '1' , the one is normally omitted. Similarly, any term raised to the power '0' equals 1.
So the answer can be simplified.
(1+3x)^4*{(1+3x) + 1}

This is the fully factored term.

Similarly:-
(3+2x)^3 + (3+2x)^4
(3+2x)^3*{1 + (3+2x)}