Factorising polynomials with complex roots

Factor theorem

Using the factor theorem, we know that if is a factor of , one of the factors of is . This also works for polynomials with complex roots.

If is a root, that means that , otherwise written as , is a factor.

Factorising polynomials with complex roots

To factorise a polynomial which has a complex root (i.e. the discriminant is negative), we can first find the complex roots by solving the polynomial equal to zero, and then use the factor theorem to write the factors.

That’s because if is a root of , then is a factor of - as we learned just above with the factor theorem.

Factorise by first solving

• Let
• Roots: or
• Factors (by the factor theorem): and
• Amnswer:

Factorise by first solving

• Let
• Roots: or
• Factors (by the factor theorem): and
• Answer:

Factorise by first solving

• Let
• Roots: or
• Factors (by the factor theorem): and
• Answer:

Factorise by first solving

• Let
• Roots: or
• Factors (by the factor theorem): and
• Answer: