In general, multiplication works with the FOIL method: Luckily, algebra with complex numbers works very predictably, here are some examples: The number a is called the real part of a+ bi, the number b is called the imaginary part of a+ bi. Let's get organized: A number of the form, where a and b are real numbers, is called a complex number. Now the polynomial has suddenly become reducible, we can write So the defining property of this imagined number i is that Here is where the mathematician steps in: She (or he) imagines that there are roots of -1 (not real numbers though) and calls them i and - i. We can't take square roots now, since the square of every real number is non-negative! Not much to complete here, transferring the constant term is all we need to do to see what the trouble is: How can we tell that the polynomial is irreducible, when we perform square-completion or use the quadratic formula? (The graph is just the standard parabola shifted up by one unit!) It cannot be factored over the real numbers, since its graph has no x-intercepts. Now you'll see mathematicians at work: making easy things harder to make them easier!Ĭonsider the polynomial. The imaginary terms of the denominator should always cancel and disappear.Factoring over the Complex Numbers Factoring over the Complex Numbers This will always happen as a result of multiplying by the conjugate. Thus, set up the example ( 4 + 3 i ) ( 2 − 2 i ). This is a useful tool for simplifying complex numbers, particularly for division problems. Whenever you multiply by a fraction whose numerator and denominator are identical, the value is just 1. Multiply the numerator and denominator by the conjugate of the denominator. The result for the sample binomial multiplication of (a+b)(c+d) is ac+ad+bc+bd. Finally, add all four products together.For the sample expression, this would be b*d. The L in FOIL represents the last terms of each binomial. In the given example, the inner terms are b*c. These would be the two terms that appear in the middle, which are the second term of the first binomial and the first term of the second binomial. The I in FOIL means to multiply the “inner” terms. These are the first term of the first binomial and the second term of the second binomial. The O in FOIL tells you to multiply the “outer” terms. The F in FOIL means that you multiply the first term of the first binomial by the first term of the second binomial. A shorthand version for doing this, is the F-O-I-L rule, which stands for “First, Outer, Inner, Last.” For an example of (a+b)(c+d), apply this rule as follows: X Research source Remember that to multiply binomials, you need to multiply each term of the first binomial by each term of the second. Looking at a complex number (a+bi) should remind you of binomials from Algebra or Algebra 2.
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