Polynomial Factoring Techniques

1. Rational Root Theorem

For a polynomial with integer coefficients:

p(x)=(a_n)*xn+(a_n−1)*x(n−1)+⋯+(a_1)*x+(a_0)

Any rational root p/q must satisfy:

• p divides (a_0) (constant term)

• q divides (a_n) (leading coefficient)

Example:

p(x)=2*x3−3*x2−8*x+12

Possible rational roots: ±{1,2,3,4,6,12}/{1,2}=±1,±2,±3,±4,±6,±12,±1/2,±3/2

Test x=2 p(2)=16−12−16+12=0 ✓

2. Synthetic Division

Once you find root r divide p(x) by (x−r)

For p(x)=2*x3−3*x2−8*x+12 with root x=2

So: p(x)=(x−2)*(2*x2+x−6)=(x−2)*(2*x−3)*(x+2)

3. Factoring by Grouping

Example:

x3+2*x2−9*x−18

Group: (x3+2*x2)+(−9*x−18)

Factor each: x2*(x+2)−9*(x+2)

Factor out (x+2) (x+2)*(x2−9)=(x+2)*(x−3)*(x+3)

4. Substitution for Even Powers

Example:

x4−5*x2+4

Let u=x2

u2−5*u+4=(u−1)*(u−4)

Substitute back:

(x2−1)*(x2−4)=(x−1)*(x+1)*(x−2)*(x+2)

5. Sum/Difference of Cubes

a3+b3=(a+b)*(a2−a*b+b2)

a3−b3=(a−b)*(a2+a*b+b2)

Example:

x3−27=x3−33=(x−3)*(x2+3*x+9)