Question

What are methods for factoring polynomials? (finding real roots, not complex ones)

I know some:
1) factor common terms
2) special products
3) factoring by grouping

Are there any other ways for factoring?

Answer 1

and ofcourse, completing the square

Answer 2

well you can do long division.

Answer 3

That is true. But you always need to find a first factor for that

Answer 4

truuueeeeeeeee

Answer 5

quadratic formula for parabola ?

Answer 6

idk cant think of more

Answer 7

True but that's the same thing as "completing the square"

Answer 8

I know there are more because my math teacher said so

Answer 9

not really. Completing the square in my opinion is a different method

Answer 10

When you complete the square on a general polynomial ax^2 + bx + c, you get the quadratic formula

Answer 11

you can factor a quadratic by inspecting the factors of numbers involved

eg
x^2 - 4x + 3
factors to (x +/- a)(x +/- b)

ab must = + 3
and a + b must = -4

so a = -1 and b = -3

and factors are (x - 1)(x - 3)

Answer 12

What is that method called? I know it but don't know it's name. Are there more methods?

Answer 13

I want to know all :p

Answer 14

for larger polynomials there the factor theorem
inspection of the coefficients of x^3 and last coefficient can help you make a good guess

eg
x^3 + 5x^2 - 2x - 15
factors are likely to be x +- 1, +/-3, +/-5 or +/- 15

check these out by substitution in the polynomial
when you have found one use long or synthetic division to find other factors

Answer 15

I know the factor theorem. But why are those factors likely?

Answer 16

can't remember the name of the above method of likely factors - there is a nme for it.
that are likely because the last term is 15

Answer 17

3*5 = 15 , 1 *15 = 15

Answer 18

oh yes - its the rational root theorem

Answer 19

you are familiar with difference of 2 squares i expect?

Answer 20

thats a special case i guess

Answer 21

yes, a binomial multiplied by it's conjugate, right?

Answer 22

yes

Answer 23

yeah, but that falls under "special product"

Answer 24

i don't know if there is a name for the factoring of quadratics (trinomials) into 2 binomials.

Answer 25

How do computers do it? What is the algorithm?

Answer 26

(I'm a coder) :p

Answer 27

no idea

Answer 28

But can you solve any polynomial with those methods? And how do you know if they are irreducible?

Answer 29

factor, not solve, sorry

Answer 30

u can factor by finding roots , like if 2 is a root then (x-2) is a factor.

Answer 31

Yes, but that's the factor theorem

Answer 32

and how do you solve an equation like
ax^16 + bx^15+cx^14+... = 0?

Answer 33

Rational Root theorem

Answer 34

Descartes Rule of Signs

Answer 35

Wolfram

Answer 36

Root finding algorithms