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Chapter 2: Problem 91

In this problem, you will explore the relationship between factoring a quadratic expression over the complex numbers and finding the zeros of the associated quadratic function. This topic will be explained in greater detail in Chapter \(3 .\) (a) Multiply \((x+i)(x-i)\) (b) What are the zeros of \(f(x)=x^{2}+1 ?\) (c) What is the relationship between your answers to parts (a) and (b)? (d) Using your answers to parts (a)-(c) as a guide, how would you factor \(x^{2}+9 ?\) (e) Using you answers to parts (a)-(d) as a guide, how would you factor \(x^{2}+c^{2},\) where \(c\) is a positive real number?

Short Answer

Expert verified
A quadratic function \(f(x) = x^{2} + c^{2}\), where \(c\) is a real positive number, can be factored into \((x+ci)(x-ci)\). The zeros of the function are \(i*c\) and \(-i*c\).

Step by step solution

01

Multiply \((x+i)(x-i)\)

This step involves simplifying the expression \((x+i)(x-i)\). The easiest way is to use the identity \(a^{2} - b^{2} = (a+b)(a-b)\) with \(a = x\) and \(b = i\). This results in \(x^{2} - i^{2}\). Because by definition, \(i^{2} = -1\), it simplifies to \(x^{2} +1\).
02

Find the zeros of \(f(x) = x^{2} + 1\)

The zero(s) of a function are the x-value(s) where the function equals zero. Suppose \(x^{2} + 1 = 0\), moving \(1\) to the other side, and taking square root on both sides, we have \(x = \pm i\). So the zeros of the function are \(+i\) and \(-i\).
03

Explain the relationship

The factors of the expression \(x^{2} + 1\) are \((x+i)\) and \((x-i)\), which are related to the zeros of the function \(f(x) = x^{2} + 1\). In general, for a quadratic equation \(a(x-h)^{2} = 0\), \(h\) is the zero of the function. Hence, factoring the quadratic expression gives the zeros of the function.
04

Factor \(x^{2} + 9\)

In light of the relationship discovered in Step 3, if asked to factor the quadratic function \(x^{2}+9\), one can recognize that it is in the form of \(x^{2} + c^{2}\) with \(c = 3\). So, similar to the earlier example, the factors will be \((x+3i)\) and \((x-3i)\).
05

Factor a general quadratic function

Following the logic from the previous steps, a general quadratic function in the form \(x^{2} + c^{2}\) where \(c\) is a positive real number can be factored into the expressions \((x+ci)\) and \((x-ci)\). This extends the relationship discovered earlier between factoring a quadratic expression and finding the zeros of the associated function.

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Key Concepts

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

Quadratic Expressions
Quadratic expressions are polynomial expressions of degree two, and they typically take the form of \(ax^2 + bx + c\). These expressions are central in algebra due to their versatility and appearance across various contexts.

What makes quadratic expressions unique is their graphing into a U-shaped curve known as a parabola. The roots or zeros of these expressions determine where the parabola intersects the x-axis. In complex numbers, quadratic expressions gain extra layers of depth.

Using complex numbers allows us to explore expressions with non-real roots. For instance, the quadratic expression \(x^2 + 1\), when graphed, does not intersect the x-axis in real coordinates. Factoring and understanding these expressions in the complex plane reveals more insights through complex roots.
Factoring
Factoring is the process of breaking down an expression into a product of simpler terms or factors. In the case of quadratic expressions, this means expressing them in the form of \((x - p)(x - q)\), where \(p\) and \(q\) are the roots.

This technique helps us not only in algebra but also in finding solutions to equations that seem unmanageable at first glance. For example, the expression \(x^2 + 9\) can be factored as \((x + 3i)(x - 3i)\). This allows us to solve it efficiently by setting each factor to zero.

The process involves several strategies including "completing the square" or using the quadratic formula, but when working with complex numbers, recognizing patterns like \(a^2 + b^2 = (a + bi)(a - bi)\) becomes essential for factorization.
Zeros of Functions
Zeros of functions, often referred to as roots, are the values that make the function equal to zero. For quadratic functions, these zeros are crucial because they describe the x-coordinates where the graph of the function will touch or cross the x-axis.

For example, the zeros of the function \(f(x) = x^2 + 1\) are \(+i\) and \(-i\). This concept is particularly important when dealing with complex numbers, because it shows how the function behaves in the complex plane鈥攂eyond the real number line.

Finding the zeros involves solving the equation \(f(x) = 0\). The relationship between factoring and finding zeros is strong: once you've factored a quadratic expression, the zeros can be directly read from the factors, indicating where \(f(x)\) equals zero.

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Most popular questions from this chapter

Let \(S(x)\) represent the weekly salary of a salesperson, where \(x\) is the weekly dollar amount of sales generated. If the salesperson pays \(15 \%\) of her salary in federal taxes, express her after-tax salary in terms of \(S(x)\) Assume there are no other deductions to her salary.

A ball is thrown directly upward from ground level at time \(t=0\) ( \(t\) is in seconds). At \(t=3,\) the ball reaches its maximum distance from the ground, which is 144 feet. Assume that the distance of the ball from the ground (in feet) at time \(t\) is given by a quadratic function \(d(t) .\) Find an expression for \(d(t)\) in the form \(d(t)=a(t-h)^{2}+k\) by performing the following steps. (a) From the given information, find the values of \(h\) and \(k\) and substitute them into the expression \(d(t)=a(t-h)^{2}+k\) (b) Now find \(a\). To do this, use the fact that at time \(t=0\) the ball is at ground level. This will give you an equation having just \(a\) as a variable. Solve for \(a\) (c) Now, substitute the value you found for \(a\) into the expression you found in part (a). (d) Check your answer. Is (3,144) the vertex of the associated parabola? Does the parabola pass through (0,0)\(?\)

Sketch a graph of the quadratic function, indicating the vertex, the axis of symmetry, and any \(x\)-intercepts. $$h(t)=-5 t+3-t^{2}$$

A rectangular plot situated along a river is to be fenced in. The side of the plot bordering the river will not need fencing. The builder has 100 feet of fencing available. (a) Write an equation relating the amount of fencing material available to the lengths of the three sides of the plot that are to be fenced. (b) Use the equation in part (a) to write an expression for the width of the enclosed region in terms of its length. (c) Write an expression for the area of the plot in terms of its length. (d) Find the dimensions that will yield the maximum area.

Is it true that \((f g)(x)\) is the same as \((f \circ g)(x)\) for any functions \(f\) and \(g ?\) Explain.
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