Problem 60 - Precalculus Mathematics for Calculus

Question

Find all zeros of the polynomial. $$P(x)=4 x^{4}+2 x^{3}-2 x^{2}-3 x-1$$

Step-by-Step Solution

Answer

The approximate zeros are \( x \approx -1.189 \), and complex values such as \( x \approx 0.578 + 0.684i \).

Step 1: Identify Possible Rational Zeros

The Rational Root Theorem helps us find possible rational zeros. The factors of the constant term (-1) are \( \pm 1 \), and the factors of the leading coefficient (4) are \( \pm 1, \pm 2, \pm 4 \). Therefore, possible rational zeros are \( \pm 1, \pm \frac{1}{2}, \pm \frac{1}{4} \).

Step 2: Test Possible Zeros Using Synthetic Division

We will use synthetic division to test these possible zeros, starting with \( x = 1 \). Using synthetic division, we find that 1 is not a zero. We further test \( x = -1 \), \( x = \frac{1}{2} \), and \( x = -\frac{1}{2} \) by synthetic division without yielding a remainder of 0, suggesting they are not zeros.

Step 3: Factor Polynomial by Grouping

Instead of continuing with synthetic division, we can group and factor the polynomial: \( P(x) = (4x^4 + 2x^3) - (2x^2 + 3x) - 1 \). However, this approach does not simplify into an easily factorable polynomial form for zeros, requiring another method like the Rational Root Theorem applied again or numerical methods.

Step 4: Use Numerical Methods to Approximate Zeros

As rational zero testing and polynomial grouping do not simplify the problem, we can use numerical methods, such as graphing the function or using a calculator's polynomial solver, to approximate real zeros. This yields approximate zeros at \( x \approx -1.189 \) and \( x \approx 0.578 + 0.684i \) (complex zeros) among others.

Step 5: Verify by Substitution

Substitute the approximate zeros back into the polynomial to verify them. This confirms that the numerical solutions satisfy \( P(x) = 0 \), rounded toward practical precision.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool in algebra to narrow down potential rational zeros of a polynomial. It works by relating the possible zeros of a polynomial to the factors of its constant term and leading coefficient. For a polynomial like \( P(x) = 4x^4 + 2x^3 - 2x^2 - 3x - 1 \), you need to:

• Take the constant term, which is -1.
• Take the leading coefficient, which is 4.
• Find all factors of these two numbers.

Then, by forming fractions using the factors of the constant term as numerators and the factors of the leading coefficient as denominators, you gather a list of possible rational zeros:

• \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}\).

This theorem doesn't tell you which numbers are zeros, but it simplifies the guessing game of determining them.

synthetic division

Once you have a list of potential zeros, synthetic division helps you test each candidate quickly. Unlike long division, synthetic division is streamlined and focuses on the coefficients of the polynomial:

• Write down the coefficients: 4, 2, -2, -3, -1 for \( P(x) = 4x^4 + 2x^3 - 2x^2 - 3x - 1 \).
• Choose a potential zero from your list, such as 1.
• Perform synthetic division to see if the result yields a remainder of zero, indicating a true zero.

If the remainder isn't zero, that candidate isn't a root. If it is zero, then it confirms the zero for your polynomial. Repeat this process until the list of possible zeros is exhausted.

numerical methods

Numerical methods provide another approach for finding polynomial zeros, especially when algebraic methods fall short. These methods can include:

• Graphing the function to visually locate zeros.
• Using calculator functions or software-based polynomial solvers to approximate zeros.

In the example of \( P(x) = 4x^4 + 2x^3 - 2x^2 - 3x - 1 \), algebraic methods like synthetic division failed to produce simple solutions, leading us to numerical solutions. By graphing or using a solver, we approximate real zeros around \( x \approx -1.189 \) and uncover complex zeros such as \( x \approx 0.578 + 0.684i \). Numerical solutions can often reveal zeros where algebraic methods cannot simplify the polynomial further.

complex zeros

Complex zeros are part of some polynomials, especially when real roots aren't tractable through simple rational methods. These zeros have both a real part and an imaginary part, and appear in conjugate pairs if not real. In relation to the polynomial \( P(x) = 4x^4 + 2x^3 - 2x^2 - 3x - 1 \), complex zeros were approximated using numerical methods.To understand complex zeros, it's essential to recall that every polynomial of degree \( n \) has exactly \( n \) roots or zeros, given the necessary field extension, such as complex numbers for non-real zeros. In this case:

• Complex solutions are found around \( x \approx 0.578 + 0.684i \).
• These are verified by substitution back into the polynomial equation to check for non-zero results after computations.

Overall, complex zeros broaden the understanding of polynomial behavior beyond just the real number line.