(–5 + 6i) – (2 + 4i)Distribute the negative sign to both terms in the second set of parentheses to get
–5 + 6i – 2 – 4iCombine like terms to get –7 + 2i, which is (A).
Since you never ended up with an i^2 term, you never even needed to worry about the fact that . You
just treat i as a regular variable.
The SAT may also test your ability to multiply complex numbers. Again you can treat i as a variable as
you work through the multiplication as if you were multiplying binominals. In other words, use FOIL to
work through the problem. The only difference is that you substitute –1 for i^2.
Finally, you may be asked about fractions with complex numbers in the denominator. Don’t worry—you
won’t need polynomial or synthetic division for this. You just need to rationalize the denominator, which
is much easier than it may sound.
To rationalize the denominator of a fraction containing complex numbers, you need to multiply the
numerator and denominator by the conjugate. To create the conjugate of a complex number, you simply
need to switch the addition or subtraction sign connecting the real and imaginary parts of the number for
its opposite.
For example, the conjugate of 8 + 7i is 8 – 7i, and the conjugate of 3 – 4i is 3 + 4i.
Just like when you expand the expression (x + y)(x − y) to get x^2 – y^2 , you can do the same with a complex
number and its conjugate. The Outer and Inner terms will cancel out, giving you (x + yi)(x − yi) = (x^2 –
i^2 y^2 ) = (x^2 + y^2 ).
(8 + 7i) × (8 – 7i) = 8^2 – 7^2 i^2 , and substituting i^2 = –1 gives you 8^2 + 7^2 = 113.
Here’s an example of how the SAT will use complex numbers in a fraction.
3.If , which of the following is equivalent to ?A) 2 +