Find the factors of x^3−7x−6

To find the factors of x³-7x-6, a method called equating coefficients can be used.

Firstly, set the equation to zero and find a suitable number which satisfies the equation, in this case x=-1 satisfies the equation, therefore one factor is (x+1), therefore the cubic equation can be represented by:

(x+1)(Ax²+Bx+C)

Expanding this polynomial gives:

x(Ax²+Bx+C)+(Ax²+Bx+C)

Ax³+Bx²+Cx+Ax²+Bx+C

Ax³+(A+B)x²+(C+B)x+C

Equating the coefficients looks at the number before the variable and comparing it with the predicted equation.

Ax³=1x³ ................................(1)

(A+B)x²=0x²...........................(2)

(C+B)x=-7x ............................(3)

C=-6 ......................................(4)

therefore A=1, C=-6, and B=-1

(x-1)(x²-x-6)

and this is a matter of factorising (x²-x-6) which is (x-3)(x-2)

and so the factorised form becomes:

(x-1)(x-3)(x-2)