)==== مثال 1: بدون باقي ====Let:formula_23Using the methods described above, the rational roots of "P"("x") are::formula_24Then, the product of ("x" − each root) is:formula_25And "P"("x")/"R"("x")::formula_26Hence the factored polynomial is "P"("x") = "R"("x") · 1 = "R"("x")::formula_27==== مثال 2: مع وجود باقي ====Let:formula_28Using the methods described above, the rational roots of "P"("x") are::formula_29Then, the product of ("x" − each root) is:formula_30And "P"("x")/"R"("x"):formula_31As formula_32, the factored polynomial is "P"("x") = "R"("x") · "S"("x")::formula_33To completely factor a given polynomial over C, the complex numbers, we must know all of its roots (and that could include irrational and/or complex numbers).
)====Example 1: no remainder====Let:formula_23Using the methods described above, the rational roots of "P"("x") are::formula_24Then, the product of ("x" − each root) is:formula_25And "P"("x")/"R"("x")::formula_26Hence the factored polynomial is "P"("x") = "R"("x") · 1 = "R"("x")::formula_27====Example 2: with remainder====Let:formula_28Using the methods described above, the rational roots of "P"("x") are::formula_29Then, the product of ("x" − each root) is:formula_30And "P"("x")/"R"("x"):formula_31As formula_32, the factored polynomial is "P"("x") = "R"("x") · "S"("x")::formula_33====Factoring over the complexes====To completely factor a given polynomial over C, the complex numbers, we must know all of its roots (and that could include irrational and/or complex numbers).