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Chapter 1: Problem 119

For Exercises \(119-124,\) factor the expressions over the set of complex numbers. For assistance, consider these examples. \(\cdot\) In Chapter \(\mathrm{R}\) we saw that some expressions factor over the set of integers. For example: \(x^{2}-4=(x+2)(x-2)\). \(\cdot\) Some expressions factor over the set of irrational numbers. For example: \(x^{2}-5=(x+\sqrt{5})(x-\sqrt{5})\). \(\cdot\) To factor an expression such as \(x^{2}+4,\) we need to factor over the set of complex numbers. For example, verify that \(x^{2}+4=(x+2 i)(x-2 i)\). a. \(x^{2}-9\) b. \(x^{2}+9\)

Short Answer

Expert verified
a. \(x^{2} - 9 = (x + 3)(x - 3)\) b. \(x^{2} + 9 = (x + 3i)(x - 3i)\)

Step by step solution

01

Identify the Type of Expression

Determine whether the expression involves a difference of squares or can be factored using complex numbers. For example, expressions like \(x^{2} - 9\) are differences of squares.
02

Factor the Difference of Squares

For the expression \(x^{2} - 9\), identify it as a difference of squares where \(a^{2} - b^{2} = (a + b)(a - b)\). Here, \(a = x\) and \(b = 3\). Therefore, \(x^{2} - 9 = (x + 3)(x - 3)\).
03

Factor Using Complex Numbers

For the expression \(x^{2} + 9\), notice that it cannot be factored using real numbers alone. We can use complex numbers where \(a^{2} + b^{2} = (a + bi)(a - bi)\). Here, \(a = x\) and \(b = 3i\). Therefore, \(x^{2} + 9 = (x + 3i)(x - 3i)\).
04

Final Expressions

Thus, the factored forms are \(x^{2} - 9 = (x + 3)(x - 3)\) and \(x^{2} + 9 = (x + 3i)(x - 3i)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The difference of squares is a special kind of polynomial that can be factored easily. The general form is \(a^2 - b^2\). It factors into two binomials: \((a + b)(a - b)\). This happens because when you expand \((a + b)(a - b)\), the middle terms cancel each other out. For example, in the expression \(x^2 - 9\), we identify that it fits the difference of squares pattern where \(a = x\) and \(b = 3\). Therefore, \(x^2 - 9\) factors into \((x + 3)(x - 3)\).
Complex Numbers
Complex numbers are numbers that have both a real part and an imaginary part. The imaginary unit is represented by \(i\), where \(i^2 = -1\). Complex numbers are used to factor equations that cannot be solved using just real numbers. For example, the expression \(x^2 + 9\) cannot be factored using real numbers. Instead, we use complex numbers: \(x^2 + 9 = (x + 3i)(x - 3i)\). Here, 3i is the imaginary part. This way, we can completely factor the polynomial.
Factoring Polynomials
Factoring polynomials involves rewriting a polynomial expression as the product of its factors. This is particularly helpful in solving polynomial equations. To factor a polynomial, you need to identify patterns or use methods like grouping or the quadratic formula. For instance, recognizing a polynomial as a difference of squares or a sum of squares can expedite the factoring process. In the exercise, we factored \(x^2 - 9\) as \((x + 3)(x - 3)\) and \(x^2 + 9\) as \((x + 3i)(x - 3i)\) by identifying the appropriate patterns.
Irrational Numbers
Irrational numbers are numbers that cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal expansions. Common examples include \(\sqrt{2}\) and \(\pi\). When factoring polynomials, some expressions may factor over the set of irrational numbers. For example, \(x^2 - 5\) can be factored as \((x + \sqrt{5})(x - \sqrt{5})\). It's crucial to recognize when an expression requires irrational numbers for factoring, as this aligns with identifying the right method to house them in the expression.

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Most popular questions from this chapter

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