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Chapter 2: Problem 47

Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) $$2,5+i$$

Short Answer

Expert verified
A real-coefficients polynomial with the given roots is \(f(x) = x^3 - 12x^2 + 50x -52\).

Step by step solution

01

Identify the Conjugate Root

Firstly, identify the conjugate root for the complex number. The conjugate of \(5+i\) is \(5-i\). Since complex roots come in pairs for polynomials with real coefficients, we add \(5-i\) to our set of roots. So, our roots are \(2\), \(5+i\), and \(5-i\).
02

Write Factors from Roots

Secondly, each root translates into a factor of the polynomial. We write factors using the form \((x-r)\), where \(r\) is the root. Our factors corresponding to the roots \(2\), \(5+i\), and \(5-i\) are \((x-2)\), \((x-(5+i))\), and \((x-(5-i))\).
03

Simplify Factors

Simplify each factor \((x-(5+i))\) and \((x-(5-i))\) for better view, getting \((x-5-i)\) and \((x-5+i)\).
04

Form the Polynomial

Finally, we form the polynomial from these factors. The polynomial \(f(x)\) we are searching for would be the multiplication of these three factors, thus \(f(x)=(x-2)(x-5-i)(x-5+i)\).
05

Expand the Polynomial

Multiply out these factors to get a polynomial: First, we multiply \((x-5-i)\) and \((x-5+i)\) getting \((x-5)^2 - i^2\) which simplifies to \(x^2 - 10x + 26\). Multiply this result with \((x-2)\) to get \(f(x) = x^3 - 12x^2 + 50x -52\)

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Most popular questions from this chapter

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x+4=0$$

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=x^{3}+24 x^{2}+214 x+740$$

The coordinate system shown below is called the complex plane. In the complex plane, the point that corresponds to the complex number \(a+b i\) is \((a, b)\) (GRAPH CANNOT COPY) Match each complex number with its corresponding point. (i) 3 (ii) \(3 i\) (iii) \(4+2 i\) (iv) \(2-2 i\) (v) \(-3+3 i\) (vi) \(-1-4 i\)

Use the given zero to find all the zeros of the function. Function \(f(x)=x^{3}+4 x^{2}+14 x+20\) Zero \(-1-3 i\)

Use Descartes's Rule of Signs to determine the possible numbers of positive and negative real zeros of the function. $$f(x)=-5 x^{3}+x^{2}-x+5$$
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