Problem 58 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The zeros of the polynomial are \(-1\), approximately \(0.541\), \(-1.080\), and \(1.539\).

Step 1: Identify and Simplify the Polynomial

We start with the polynomial \(P(x) = x^4 - x^2 + 2x + 2\). Our goal is to find the values of \(x\) that make the polynomial equal to zero, i.e., \(P(x) = 0\).

Step 2: Check for Rational Roots

Apply the Rational Root Theorem, which states that possible rational roots are factors of the constant term 2. This gives us potential roots: \(\pm 1\), \(\pm 2\).

Step 3: Evaluate Polynomial at Possible Roots

Substitute the possible rational roots into \(P(x)\):- \(P(1) = 1^4 - 1^2 + 2 \times 1 + 2 = 4\) (not zero)- \(P(-1) = (-1)^4 - (-1)^2 + 2 \times (-1) + 2 = 0\) (root found)- For \(x = 2\): \(P(2) = 2^4 - 2^2 + 2 \times 2 + 2 = 18\)- For \(x = -2\): \(P(-2) = (-2)^4 - (-2)^2 + 2 \times (-2) + 2 = 14\) (not zero)

Step 4: Factor the Polynomial Using Found Root

Since \(x = -1\) is a root, we can divide \(P(x)\) by \(x + 1\). Performing synthetic division or polynomial division to find \((x^4 - x^2 + 2x + 2) \div (x + 1)\) gives us a quotient of \(x^3 - x^2 + x + 2\).

Step 5: Solve the Cubic Function

Now solve \(x^3 - x^2 + x + 2 = 0\). Check for any rational roots among \(\pm 1\), \(\pm 2\). Evaluating, we find:- \(P(1) = 3\) (not zero)- \(P(-1) = 0\) (already found)- For \(x = 2\): \(P(2) = 12\)- For \(x = -2\): \(P(-2) = -4\)None work, so solve using methods such as graphing or numerical estimation.

Step 6: Use Numerical or Graphical Methods for Remaining Roots

At this point, we need to use a graphing calculator or numerical methods to approximate the remaining roots for \(x\) in the cubic equation \(x^3 - x^2 + x + 2 = 0\). Approximate roots are found to be \(x \approx 0.541\), \(x \approx -1.080\), \(x \approx 1.539\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool for finding possible rational solutions to a polynomial equation. It tells us that any potential rational root of a polynomial, with integer coefficients, will be a fraction \( \frac{p}{q} \), where:

• \( p \) is a factor of the constant term of the polynomial
• \( q \) is a factor of the leading coefficient

In our original polynomial, \( P(x) = x^4 - x^2 + 2x + 2 \), the constant term is 2, and the leading coefficient is 1. Hence, potential rational roots are the factors of 2, which are \( \pm 1 \) and \( \pm 2 \).
By substituting these values into the polynomial, we test which, if any, satisfy the equation \( P(x) = 0 \). As found, \( x = -1 \) is a valid root.

Synthetic Division

Once a potential root is confirmed, synthetic division helps to simplify the polynomial, leading us to a lower-degree equation to solve. Synthetic division is a quick method to divide a polynomial by a linear factor like \( (x - c) \). It involves using the root \( c \) and performing operations on the coefficients of the polynomial.Here's how it works:

• Write the coefficients of the polynomial in a row.
• Use the found root (for instance, \( x = -1 \) means using \( -1 \) in synthetic division).
• Carry out the multiplication and addition steps to complete the division.

For our exercise, synthetic division of \( P(x) = x^4 - x^2 + 2x + 2 \) by \( x + 1 \) results in \( x^3 - x^2 + x + 2 \).
This simplified polynomial helps us search for additional roots.

Cubic Equation

A cubic equation is a polynomial of degree three and can be more complex to solve than quadratic equations. The general form is \( ax^3 + bx^2 + cx + d = 0 \).
In our simplified problem, we reached the cubic equation \( x^3 - x^2 + x + 2 = 0 \).Finding the roots of a cubic equation involves similar strategies:

• Start with the Rational Root Theorem to check possible rational solutions.
• Factor if any roots are found, as in our prior step where no additional rational roots exist.
• Explore numerical methods or graphing to find approximate solutions if necessary.

Cubic equations often don't neatly factor and require innovative approaches for solving, like graphically or using calculus-based methods for approximations.

Numerical Approximation

Numerical approximation techniques are sometimes needed when exact roots are difficult to determine algebraically, especially in cubic equations. Methods such as graphing calculators, iterative algorithms, or even the Newton-Raphson method can approximate roots to a desired accuracy.Why use numerical approximations?

• When simple factoring doesn't work, especially for higher-degree polynomials.
• To find decimal values of roots when only an estimate is applicable.
• Applicable in complex real-world problems, like engineering or physics, where precise values are needed.

For \( x^3 - x^2 + x + 2 = 0 \), we used numerical methods, uncovering roots at approximately \( x \approx 0.541 \), \( x \approx -1.080 \), and \( x \approx 1.539 \).
This method broadens our ability to solve polynomials beyond simple algebraic methods.