Please explain for notes to take. That way I can learn **Problem 82:**There is exactly one polynomial with integer coefficients with zeros of 2, 4, and 6.**Explanation:**To find a polynomial with given zeros, we can use the fact that if a polynomial has zeros at $ x = a $, $ x = b $, and $ x = c $, then the polynomial can be expressed in the factored form:\[P(x) = k(x - a)(x - b)(x - c)\]where $ k $ is a nonzero constant.For the zeros 2, 4, and 6, and since we want integer coefficients, we can set $ k = 1 $. Therefore, the polynomial is:\[P(x) = (x - 2)(x - 4)(x - 6)\]Expanding this, we get the polynomial:\[P(x) = (x - 2)(x^2 - 10x + 24) = x^3 - 10x^2 + 24x - 2x^2 + 20x - 48\]Simplifying, the polynomial with integer coefficients is:\[P(x) = x^3 - 12x^2 + 44x - 48\]Thus, the polynomial $ P(x) = x^3 - 12x^2 + 44x - 48 $ is the polynomial with integer coefficients that has zeros at 2, 4, and 6.

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82. There is exactly one polynomial with integer coefficients with zeros of 2, 4, and 6.

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Please explain for notes to take. That way I can learn

**Problem 82:**

There is exactly one polynomial with integer coefficients with zeros of 2, 4, and 6.

**Explanation:**

To find a polynomial with given zeros, we can use the fact that if a polynomial has zeros at $ x = a $, $ x = b $, and $ x = c $, then the polynomial can be expressed in the factored form:

\[
P(x) = k(x - a)(x - b)(x - c)
\]

where $ k $ is a nonzero constant.

For the zeros 2, 4, and 6, and since we want integer coefficients, we can set $ k = 1 $. Therefore, the polynomial is:

\[
P(x) = (x - 2)(x - 4)(x - 6)
\]

Expanding this, we get the polynomial:

\[
P(x) = (x - 2)(x^2 - 10x + 24) = x^3 - 10x^2 + 24x - 2x^2 + 20x - 48
\]

Simplifying, the polynomial with integer coefficients is:

\[
P(x) = x^3 - 12x^2 + 44x - 48
\]

Thus, the polynomial $ P(x) = x^3 - 12x^2 + 44x - 48 $ is the polynomial with integer coefficients that has zeros at 2, 4, and 6.

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