# Factoring polynomial

#### bergausstein

##### Active member
any hints on how to start this problem?

$12x^4+19x^3-26x^2-61x-28$

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#### Opalg

##### MHB Oldtimer
Staff member
any hints on how to start this problem?

$12x^2+19x^3-26x^2-61x-28$
Is that $12x^2$ perhaps a typo for $12x^4$?

#### bergausstein

##### Active member
yes that's 12x^4. sorry.

#### Opalg

##### MHB Oldtimer
Staff member
Start by looking for integer roots of the polynomial (factors of the constant term). If you find any, then the factor theorem gives you linear divisors of the polynomial.

#### Deveno

##### Well-known member
MHB Math Scholar
Start by looking for integer roots of the polynomial (factors of the constant term). If you find any, then the factor theorem gives you linear divisors of the polynomial.

The roots may not be INTEGERS, as the leading term's coefficient is not 1.....

#### LATEBLOOMER

##### New member
i will use dorobostikerlines method.,

$12(x^2-1)^2+19(x^2-1)(x+1)-21(x+1)^2$

i'll let you continue.

#### Opalg

##### MHB Oldtimer
Staff member
The roots may not be INTEGERS, as the leading term's coefficient is not 1.....
True, but I like an easy life, so I look for the simplest possible solutions first.

#### soroban

##### Well-known member
Hello, bergausstein!

Any hints on how to start this problem?

$\text{Factor: }\:f(x) \:=\:12x^4+19x^3-26x^2-61x-28$

$$\text{We find that }f(\text{-}1) \,=\,0.$$
$$\text{Hence, }x+1\text{ is a factor.}$$

$$\text{Long division: }\:f(x) \:=\: (x+1)\underbrace{(12x^3 + 7x^2 - 33x - 28)}_{g(x)}$$
$$\text{We find that }g(\text{-}1) \,=\,0.$$
$$\text{Hence, }x+1\text{ is a factor.}$$

$$\text{Long division: }\:g(x) \:=\: (x+1)(12x^2-5x - 28)$$

$$\text{Can you finish it?}$$

#### topsquark

##### Well-known member
MHB Math Helper
i will use dorobostikerlines method.,

$12(x^2-1)^2+19(x^2-1)(x+1)-21(x+1)^2$

i'll let you continue.
I've never heard of this method and google comes up with nothing. Can you give us a quick run-down?

-Dan

#### bergausstein

##### Active member
latebloomer, that method seems to work correctly. and i got the right answer. can you show me how that method work in its full glory?

#### LATEBLOOMER

##### New member
actually you won't like it if I show you the full workings of this method. that method is from an indian mathematician named doroboski. well, his name was not celebrated as other great mathematicians out there so you'll rarely find information about him. i would prefer using the other method metioned above.