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Chapter 2: Problem 27

Divide and express the result in standard form. $$\frac{2+3 i}{2+i}$$

Short Answer

Expert verified
The result of the complex division is \(1.4 + 0.8i\).

Step by step solution

01

Identify the conjugate of the denominator

Start by identifying the conjugate of the denominator, which is \(2 - i\). The conjugate of a complex number changes the sign of the imaginary part. If the complex number is \(a + bi\), its conjugate is \(a - bi\).
02

Multiply the fraction by the conjugate

Multiply the numerator and the denominator of the fraction by the complex conjugate of the denominator. The resulting fraction is \[\frac{(2 + 3i)(2 - i)}{(2 + i)(2 - i)}\]
03

Apply the FOIL method

Apply the FOIL method (First, Outer, Inner, Last) to the expressions in the numerator and the denominator. \[\frac{4 - 2i + 6i - 3i^2}{4 - 2i + 2i - i^2}\]. Remember that \(i^2\) equals -1.
04

Simplify the fraction

Simplify the fraction to get the real and imaginary parts. \[\frac{4 + 4i + 3}{4 + 1} = \frac{7 + 4i}{5}\]
05

Write out the Final Answer in Standard Form

Finally, write the answer in standard form (a + bi) by splitting the fraction. It becomes \(1.4 + 0.8i\)

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Most popular questions from this chapter

If \(f\) is a polynomial or rational function, explain how the graph of \(f\) can be used to visualize the solution set of the inequality \(f(x)<0\).

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations of \(f(x)=\frac{1}{x}\) to graph \(g.\) $$g(x)=\frac{3 x+7}{x+2}$$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$\frac{2}{x^{2}+3 x+2}-\frac{4}{x^{2}+4 x+3}$$

Find the horizontal asymptote, if there is one, of the graph of rational function. $$h(x)=\frac{15 x^{3}}{3 x^{2}+1}$$

Will help you prepare for the material covered in the next section. Simplify: \(\frac{x+1}{x+3}-2\)
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