How to factor a 5-term polynomial (the double-cross method)

Check out different methods of factoring this 5-term polynomial below.

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#math #algebra #mathbasics

[Iomhar]

6a+2b=-11
Lef side is even, right side is odd!
No integer solution!

[nevoitzhak]

The sum of the coefficients is 0 therefore 1 is a solution
X⁴-4x³+2x²-11x+12=
X⁴-x³-3x³+3x²-x²+x-12x+12=

(X-1)(x³-3x²-x-12)
Then factoring a cubic is much easier

[JoseAntonio-ngyu]

If the independent term has too many factors, this can be quite hard, as you must check every combination of two factors 2 times each, as they can also have switched signs.
The solution of the system of equations is a=(Ac-C)/(c-d) and b=(C-Ad)/(c-d). A and C are the coefficients of x^3 and x, and c and d are the top and bottom numbers at the right cross which are factors of the independent term. Substitute directly into a to eliminate more quickly those in which a is not a in integer. If a is an integer, obtain b and check the last condition

[a_man]

My solution: first factor out x^3 from first two terms, then factorize remaining second degree polynomial, then factor out (x-4), i found the x-1 factor by trying in my head

[major__kong]

(x - r1)(x - r2)(x - r3)(x - r4) = ax^4 + bx^3 + cx^2 + dx + e

Multiply out the left side, set the coefficients of like powers of x equal to each other, solve the system of equations for a, b, c, d, and e in terms of r1, r2, r3, and r4. Hint: a = 1. :-)

[giorgouis]

could you do the Δ(4th/D) problem from the Greek panhellenics 2024 if possible?

[lushleafy]

this is harder than the synthetic division method, i don't this i can use this method efficiently, i am too dumb

[Ben_Long]

Do you have a video on lots of different ways to factor different polynomials?

[kadinatorgaming]

Please help me factor 5x^4-7x^3+3x^2-x+1

[dutchie]

2:30 why do a and b need to be whole numbers? Is there a rule for that?

[aneeshbro]

bro when are u gonna solve my problem? I need the answer, u explain everything in the best way possible.

[wdobni]

you should be able to calculate whether useful quantum computers will ever be developed....the limiting factor is noise and we know how much noise there is and how much each unit of noise reduces quantum integrity with respect to should be a straightforward matter to compute whether noise will ever be sufficiently mastered to permit reliable useful quantum computing beyond the toy stage