# What are the factors in algebra

## Meinstein

Factoring or factorization of polynomials in algebra is understood as the decomposition of polynomials into a product of polynomials (expressions) that can no longer be decomposed, just like the prime factorization of whole numbers.

Work according to the following grid:

• Can a common factor be written in front of the brackets?
• Is it a binomial formula?
• Is it a binomial-like formula (3 terms, one square)?
• Can you make progress with forming a group (often a sum of four summands)?
• Am I done or can a term be factored further?

### Examples

1.

m (r - s) - n (s - r) =
m (r - s) + n (r - s) = we multiply the second bracket by -1
(r - s) (m + n) we exclude.

2.

-4s + 8t + t - 10s - 5t =
s (- 4 - 10) + t (8 + 1 - 5 =
- 14s + 4t

### Exercises

1. 24a4 - 32a3 =
2. 39a2n2 - 26an =
3. −20m + 12n - 4q =
4. 10am - 6an - 2ap =
5. 7a2b - 21ab2 + from =
6. - ac - bc - c =
7. y3 - y2 =
8. 2a3bc + 8a2b2c - 2ab3c - 2a2bc2 + 16abc3 =
9. −6x4y4z4 + 18x3y3z3 - 12x2y2z3 =
10. 36m5n6 - 90m4n7 - 180m3n8 =

Solutions:

8a3 (3a - 4)

13an (3an - 2)

- 4 (5m - 3n + q)
It is better here to factor out -4; Be careful with the signs!

2a (5m - 3n - p)

Don't forget the 1!

from (7a - 21b + 1)

c (a + b + 1)

y2 (y - 1)

Perhaps as a precaution, write the terms one below the other:
2abc (a2 + 4ab - b2 - ac + 8c2)

2a3
bc + 8a2b2c - 2ab3
c - 2a2
bc2 + 16abc3 =
2abc (a2 + 4ab - b2 - ac + 8c2)
9
Be selective about the term that you put in front of the parentheses:
first consider only the existing numbers, then the x, then the y, then the z.
−6x 4y4z4 + 18 × 3 y3 z3 - 12x2y2z3 = −6 × 2 y2 z3 (x2 y2 z - 3xy + 2)
10 36m5n6 - 90m4n7 - 180m3n8 = 18m3n6 (2m2 - 5mn - 10n2)

### Similar issues

Prime numbers

Prime factorization

Factoring trinomials