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Chapter 3: Problem 97

Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary. $$f(x)=x^{5}-6 x^{4}+14 x^{3}-20 x^{2}+24 x-16$$

Short Answer

Expert verified
Zeros: \( x = 2 \) (multiplicity 3), \( x = i\sqrt{2} \), \( x = -i\sqrt{2} \)

Step by step solution

01

Identify Rational Roots

Use the Rational Root Theorem to identify possible rational roots. The potential rational roots are the factors of the constant term divided by the factors of the leading coefficient. For the polynomial\( f(x) = x^5 - 6x^4 + 14x^3 - 20x^2 + 24x - 16 \), these roots are \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 \).
02

Test Rational Roots Using Synthetic Division

Use synthetic division to test these potential roots. One correctly identifies \( x = 2 \) as a root. Dividing \( f(x) \) by \( x - 2 \) yields \( x^5 - 6x^4 + 14x^3 - 20x^2 + 24x - 16 = (x - 2)(x^4 - 4x^3 + 6x^2 - 8x + 8) \).
03

Factor the Quotient Polynomial

Continuing the process with \( x^4 - 4x^3 + 6x^2 - 8x + 8 \), use the same method to find other rational roots. Test and identify the next root, \( x = 2 \). Repeat the division to obtain\( x^4 - 4x^3 + 6x^2 - 8x + 8 = (x - 2)^2 (x^3 - 2x^2 + 2x - 4) \).
04

Continue Factoring

Now, factor \( x^3 - 2x^2 + 2x - 4 \) using rational root testing. Test and find \( x = 2 \) once more. Dividing results in\( x^3 - 2x^2 + 2x - 4 = (x - 2)^3(x^2 + 2) \).
05

Find Complex Roots

Finally, solve \( x^2 + 2 \) to find the complex roots. The solutions to \( x^2 + 2 = 0 \) are \( x = \pm i\sqrt{2} \). This gives the remaining complex zeros \( x = i\sqrt{2} \) and \( x = -i\sqrt{2} \).
06

List All Zeros

Summarize the zeroes discovered. The polynomial \( f(x) \) has the zeros \( x = 2 \) with multiplicity 3, and \( x = i\sqrt{2} \) and \( x = -i\sqrt{2} \).

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Key Concepts

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

Rational Root Theorem
The Rational Root Theorem is a useful tool in finding the possible rational roots of a polynomial. It states that any rational root of a polynomial equation, where the polynomial is given as \[ P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \], must be a fraction \( \frac{p}{q} \) where:
  • \( p \) is a factor of the constant term \( a_0 \)
  • \( q \) is a factor of the leading coefficient \( a_n \)
. For instance, in the polynomial \( f(x)=x^5-6x^4+14x^3-20x^2+24x-16 \), the Rational Root Theorem allows us to list the possible roots as \( \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 \). This set of numbers forms the pool of candidates we will test to find actual roots.
Synthetic Division
Synthetic division simplifies the process of polynomial division, especially when trying to find roots. Instead of traditional long division, synthetic division uses a streamlined approach to test possible roots. Given a candidate root \( r \), we set up a synthetic division table, and through iterative steps, we determine if \( r \) is indeed a root. For example, testing \( x = 2 \) for \( f(x)=x^5-6x^4+14x^3-20x^2+24x-16 \) involves synthetic division, which confirms 2 as a root and allows us to factor the polynomial further.
Multiplicity of Zeros
Zeros of a polynomial can have different multiplicities. The multiplicity of a zero refers to how many times it appears as a root. For instance, in \( f(x) \), we found \( x=2 \) multiple times during our factorization process. Specifically, \( x=2 \) appeared three times, giving it a multiplicity of 3. Thus, the zero \( x=2 \) is said to have a multiplicity of 3.
Complex Roots
Complex roots arise when solving polynomial equations involving the square roots of negative numbers. When factoring polynomials, especially quadratic components, we can encounter terms that lead to complex numbers. For example, in our solution, after reducing the polynomial, we ended up with \( x^2 + 2 \). Solving \( x^2 + 2 = 0 \) involves introducing the imaginary unit \( i \), where \( i = \sqrt{-1} \). Thus, the roots become \( x = \pm i\sqrt{2} \), which are the complex roots for our polynomial.
Factoring Polynomials
Factoring is the process of breaking down a polynomial into simpler components ('factors') that, when multiplied together, give back the original polynomial. To find the zeros of a polynomial, we repeatedly use methods like the Rational Root Theorem and synthetic division to decompose the polynomial. For \( f(x)=x^5-6x^4+14x^3-20x^2+24x-16 \), we factor it first to \( (x-2)(x^4-4x^3+6x^2-8x+8) \), then further to \( (x-2)^3(x^2+2) \). Factoring continues until we have completely broken down the polynomial into irreducible components, helping us identify all the roots, real and complex.

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