3.1.2—Polynomial division

Use a known root a to divide P(x) by x−a, obtaining a lower-degree quotient. Then factor or solve the quotient and retain only roots satisfying the original polynomial.

Use the remainder theorem to test candidates, include missing powers as zero coefficients, and verify by multiplication. Repeated division reveals multiplicity.

If P(−1)=0 for a quartic, divide by x+1; if the quotient also vanishes at −1, the factor is repeated.

A factor found by trial does not guarantee the remaining quotient is quadratic; degree bookkeeping must be preserved.