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Chapter 2: Problem 18

Find each product and write the result in standard form. $$(-7-i)(-7+i)$$

Short Answer

Expert verified
The product of the complex numbers (-7 - i) and (-7 + i) in standard form is \(48 - 14i\).

Step by step solution

01

Apply the distributive property

A product of two binomials can be calculated using the distributive property, also referred to as FOIL. This method requires multiplying the first terms, the outer terms, the inner terms, and the last terms of the binomials, respectively. Thus, \[(-7 - i) (-7 + i) = (-7*-7) + (-7*i) + (-7*i) + ((-1*i)*i)\]
02

Simplify Each Product

After applying FOIL, you must simplify each product: \[49 - 7i - 7i + (1*i^2)\]
03

Use \(i^2 = -1\)

It is crucial to remember the definition of \(i\). Simplify the term including \(i^2\), with the understanding that \(i^2 = -1\). Thus, it implies: \[49 - 7i - 7i - 1.\]
04

Group like terms

Combine the real parts (49 and -1) and the imaginary parts (-7i and -7i) separately to simplify the expression, resulting to: \[48 - 14i .\]

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \(x+3,\) resulting in the equivalent inequality \(x-2<2(x+3)\).

What is a polynomial inequality?

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, I only pay attention to the sign of \(f\) at each test value and not the actual function value.

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