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Chapter 3: Problem 81

Write a polynomial \(f(x)\) that meets the given conditions. Answers may vary. (See Example 10 ) Degree 2 polynomial with zeros of \(7+8 i\) and \(7-8 i\).

Short Answer

Expert verified
The polynomial is \( f(x) = x^2 - 14x + 113 \).

Step by step solution

01

Understand the Problem

It is given that the polynomial should be of degree 2 and have zeros at 7+8i and 7-8i. For a degree 2 polynomial, the general form is: \(f(x) = a(x-r_1)(x-r_2)\), where \(r_1\) and \(r_2\) are the zeros.
02

Write the Polynomial Using Given Zeros

Substitute the given zeros \(7+8i\) and \(7-8i\) into the polynomial form: \(f(x) = a(x-(7+8i))(x-(7-8i))\).
03

Simplify the Expression

Simplify the expression. First subtract the complex numbers inside the parentheses: \(f(x) = a((x-7-8i)(x-7+8i))\).
04

Apply Difference of Squares

Use the difference of squares formula, \((x-a)(x+b) = x^2-(b^2)\), to simplify further: \(f(x) = a((x-7)^2-(8i)^2)\).
05

Simplify Complex Numbers

Simplify the complex number term \( (8i)^2 = -64 \): \(f(x) = a((x-7)^2-(-64))\).
06

Combine and Expand

Combine terms and expand \((x-7)^2 - (-64) = (x-7)^2 + 64\). Expand \((x-7)^2 = x^2 - 14x + 49\): \( f(x) = a (x^2 - 14x + 49 + 64 )\). Simplify further: \( f(x) = a (x^2 - 14x + 113)\).
07

Determine the Leading Coefficient

Choose the leading coefficient, \(a\), to be 1 for simplicity: \( f(x) = x^2 - 14x + 113\).

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Key Concepts

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

Zeros of Polynomial
Zer芯s of a polyn芯mi邪l refer to th械 values 芯f x for which the polyn芯mi邪l evaluat械s to zero. Essentially, these are the solutions or roots of the polynomial equation. For instance, if the polynomial is given by \(f(x) = x^2 - 14x + 113\), the zeros are found by solving the equation \(f(x) = 0\). In this case, the given values are complex numbers, \(7+8i\) and \(7-8i\). Here鈥檚 how this works:
  • Each zero represents a solution for \(x\) in the polynomial equation \(f(x) = 0\).
  • A polynomial of degree \(n\) has exactly \(n\) zeros, which may include real and complex numbers.
Understanding these zeros helps in constructing the polynomial in its factored form. By substituting the zeros into the general polynomial form, we can create the required polynomial.
Degree of Polynomial
The degree of a polynomial refers to the highest power of the variable \(x\) in the polynomial expression. It is a crucial concept since it determines the number of zeros a polynomial can have and the general behavior of the polynomial at very large or very small values of \(x\). For example, in the polynomial \(f(x) = x^2 - 14x + 113\), the highest power of \(x\) is \(2\), making this a degree \(2\) polynomial. Here鈥檚 why it matters:
  • A degree 2 polynomial, also called a quadratic polynomial, will have up to 2 zeros.
  • The shape and characteristics of the polynomial graph are heavily influenced by its degree.
In our exercise, since we needed a degree 2 polynomial, it naturally led us to consider a quadratic form to fit the given zeros.
Complex Numbers
Complex numbers are numbers that include an imaginary component along with a real component and are usually written in the form \(a + bi\) where \(i\) is the imaginary unit satisfying \(i^2 = -1\). The given exercise involved zeros that were complex numbers: \(7+8i\) and \(7-8i\). Here鈥檚 an overview of the complex numbers:
  • Real Part: The real component, denoted as \(a\) in \(a+bi\), is a standard real number.
  • Imaginary Part: The imaginary component, denoted as \(bi\), includes the imaginary unit \(i\) which squared equals \(-1\).
  • Conjugates: Complex conjugates are pairs like \(7+8i\) and \(7-8i\). They have the same real part but opposite imaginary parts.
In the context of polynomials, complex zeros typically appear in conjugate pairs when the polynomial has real coefficients. This fact greatly simplifies the factoring and construction of the polynomial.

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Most popular questions from this chapter

Determine if the statement is true or false. a. Use the quadratic formula to solve \(x^{2}-7 x+5=0\) b. Write \(x^{2}-7 x+5\) as a product of linear factors.

Suppose that an object that is originally at room temperature of \(32^{\circ} \mathrm{C}\) is placed in a freezer. The temperature \(T(x)\) (in \({ }^{\circ} \mathrm{C}\) ) of the object can be approximated by the model \(T(x)=\frac{320}{x^{2}+3 x+10}\), where \(x\) is the time in hours after the object is placed in the freezer. a. What is the horizontal asymptote of the graph of this function and what does it represent in the context of this problem? b. A chemist needs a compound cooled to less than \(5^{\circ} \mathrm{C}\). Determine the amount of time required for the compound to cool so that its temperature is less than \(5^{\circ} \mathrm{C}\).

Suppose that \(y\) varies jointly as \(x\) and \(w^{3}\). If \(x\) is replaced by \(\frac{1}{3} x\) and \(w\) is replaced by \(3 w,\) what is the effect on \(y\) ?

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ \sqrt{2 x-6}-2<0 $$

The procedure to solve a polynomial or rational inequality may be applied to all inequalities of the form \(f(x)>0, f(x)<0,\) \(f(x) \geq 0,\) and \(f(x) \leq 0 .\) That is, find the real solutions to the related equation and determine restricted values of \(x .\) Then determine the sign of \(f(x)\) on each interval defined by the boundary points. Use this process to solve the inequalities. $$ 8 \leq x^{2}+4 x+3<15 $$
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