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Chapter 5: Problem 41

Find the complex zeros of each polynomial function. Write fin factored form. $$ f(x)=3 x^{4}-x^{3}-9 x^{2}+159 x-52 $$

Short Answer

Expert verified
The complex zeros of the polynomial are: -4, 0.3027, 4.1532i, -4.1532i. The factored form is: 3(x + 4)(x - 0.3027)(x - 4.1532i)(x + 4.1532i).

Step by step solution

01

- Identify the polynomial and its degree

The given polynomial function is: f(x) = 3x^4 - x^3 - 9x^2 + 159x - 52 The degree of this polynomial is 4, which means there are 4 complex roots (including multiplicities).
02

- Use the Rational Root Theorem

According to the Rational Root Theorem, any rational root of the polynomial is a factor of the constant term (-52) divided by a factor of the leading coefficient (3). The factors of -52 are: 卤1, 卤2, 卤4, 卤13, 卤26, 卤52 The factors of 3 are: 卤1, 卤3 Therefore, the possible rational roots are: 卤1, 卤2, 卤4, 卤13, 卤26, 卤52, 卤1/3, 卤2/3, 卤4/3, 卤13/3, 卤26/3, 卤52/3
03

- Test possible rational roots

Test each possible rational root by substituting it into the polynomial and checking if f(x) = 0. Using synthetic division or direct substitution, we find that one of the roots is:x = -4
04

- Perform synthetic division

Using synthetic division, divide the polynomial 3x^4 - x^3 - 9x^2 + 159x - 52 by (x + 4): The resulting polynomial is:3x^3 - 13x^2 + 43x - 13
05

- Find the remaining roots

Repeat the previous steps to find the remaining zeros of the polynomial. Using the quadratic formula for 3x^3 - 13x^2 + 43x - 13: The remaining roots are: x 鈮� 0.3027, x 鈮� 4.1532i, x 鈮� -4.1532i
06

- Write the polynomial in factored form

Combine all the roots (including the complex roots) and write the polynomial in its factored form: f(x) = 3(x + 4)(x - 0.3027)(x - 4.1532i)(x + 4.1532i)

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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 powerful tool for identifying potential rational roots of a polynomial. Here's how it works:
The theorem states that any possible rational root of a polynomial equation \(a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 = 0\) is a fraction \(p/q\), where:\
- \(p\) is a factor of the constant term \(a_0\)
- \(q\) is a factor of the leading coefficient \(a_n\)
To apply this, you need to:
  • Identify the constant term and leading coefficient of your polynomial
  • List all factors of the constant term and leading coefficient
  • Form all possible fractions \(p/q\), including their positive and negative values
For our example polynomial \(f(x) = 3x^4 - x^3 - 9x^2 + 159x - 52\):
- The constant term is \(-52\)
- The leading coefficient is \(3\)
The possible rational roots are \(\text{卤1, 卤2, 卤4, 卤13, 卤26, 卤52}\) and \(\text{卤1/3, 卤2/3, 卤4/3, 卤13/3, 卤26/3, 卤52/3}\).
Synthetic Division
Synthetic Division is a simplified form of polynomial division, particularly useful for evaluating potential roots found using the Rational Root Theorem.
Here's a step-by-step guide to performing synthetic division:
  • Set up the synthetic division: Write down the coefficients of the polynomial in descending order of the powers of \(x\). If any powers of \(x\) are missing, use zero as their coefficient.
  • Choose a candidate root: Use one of the potential rational roots you found.
  • Perform synthetic division steps:
    • Write the candidate root on the left.
    • Bring down the leading coefficient to the bottom row.
    • Multiply the candidate root by the number you just brought down, and place the result in the next column.
    • Add this result to the coefficient above it and write the sum underneath it.
    • Repeat this process until you reach the end of the row.
If the last number (remainder) is zero, your candidate root is indeed a root of the polynomial. In our example, by using the synthetic division on \(-4\), we get a quotient polynomial \(3x^3 - 13x^2 + 43x - 13\).
Factored Form
Factoring a polynomial means expressing it as a product of its roots. Once we have identified all the roots, including complex roots, we can write the polynomial in factored form.

For the polynomial \(f(x) = 3x^4 - x^3 - 9x^2 + 159x - 52\), we found roots \(-4\), \(\tilde{x} \approx 0.3027\), \(\tilde{z} \approx \text{卤4.1532i}\).
  • Combine these roots into linear factors:
    \(f(x) = 3(x + 4)(x - 0.3027)(x - 4.1532i)(x + 4.1532i)\)
  • The corresponding terms are:
    • \(x + 4\) for root \(-4\)
    • \(x - 0.3027\) for root \(\tilde{x} \approx 0.3027\)
    • \(x - 4.1532i\) and \(x + 4.1532i\) for roots \(\text{卤4.1532i}\)
Once you multiply all these factors together, you obtain the original polynomial, now in its factored form. This process simplifies polynomial equations and helps understand their behavior and properties.

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