We can extend our knowledge of complex numbers and polynomial identities to find complex factors of polynomials.
With this type of factorization, the key is to introduce the term $i^2=-1$i2=−1 in the given expression whenever required. Let's work through a few examples.
Factor the expression $x^2+9$x2+9.
Think: The expression $x^2+9$x2+9 can only be factored if we rewrite the second term using the fact that $i^2=-1$i2=−1. Once it is rewritten, we can apply the difference of two squares polynomial identity.
Rewrite the second term using $i^2=-1$i2=−1
Factor using the difference of two squares polynomial identity
Factor the expression $4x^2-12ix-9$4x2−12ix−9.
Think: The middle term has a coefficient of $-12i$−12i. If we rewrite the last term using $i^2=-1$i2=−1, we might be able to factor the expression.
Using $i^2=-1$i2=−1, we get $-9=+9i^2$−9=+9i2
Now, factor using the identity $\left(A-B\right)^2=A^2-2AB+B^2$(A−B)2=A2−2AB+B2.
Complete the factoring by filling in the empty box.
Factor the expression $3x^2+108$3x2+108. Leave your answer in terms of $i$i.
Factor the expression $x^2+12ix-36$x2+12ix−36.
Extend polynomial identities to include factoring with complex numbers. For example, rewrite x^2 + 4 as (x + 2)(x - 2i)