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Chapter 4: Q. 25 (page 245)

Information is given about a complex polynomial function f(x) whose coefficients are real numbers. Find the remaining zeros of f. Then find a polynomial function with real coefficients that has the zeros.

Degree 3, Zeroes:4+i,6

Short Answer

Expert verified

The remaining zero is4-iand the polynomial isx3-14x2+65x-102

Step by step solution

01

Step 1. Given Information

The given data is Degree 3, Zeroes:4+i,6

02

Step 2. Explanation

For a complex polynomial function with real coefficients, if zero is a complex number then its conjugate will also be a zero of the function.

Since 4+iis complex, its conjugate will be the remaining zero.

4+i=4-i

Thus,4-iis the remaining zero.

03

Step 3. Calculation

Use the distributive property to find the polynomial using the factors.

f(x)=(x-6)[x-(4+i)][x-(4-i)]=(x-6)[x2-x(4+i)-x(4-i)+(4+i)(4-i)]=(x-6)(x2-8x+17)=(x3-8x2+17x-6x2+48x-102)=x3-14x2+65x-102

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