Problem 44 - Algebra and Trigonometry

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=x^{4}+2 x^{3}-2 x^{2}-3 x+2 $$

Step-by-Step Solution

Answer

The real zeros are \( x = 1, -1, -1 + \sqrt{3}, -1 - \sqrt{3} \).

Step 1: Check for Rational Roots Using the Rational Root Theorem

The Rational Root Theorem helps us check for potential rational roots of the polynomial. These roots must be among the factors of the constant term (2) divided by the leading coefficient (1). Thus, the possible rational roots are \( \pm 1, \pm 2 \).

Step 2: Test Possible Rational Roots

Test each possible rational root by substituting into \( P(x) \). - \( P(1) = 1^4 + 2 \cdot 1^3 - 2 \cdot 1^2 - 3 \cdot 1 + 2 = 0 \), so 1 is a root.- \( P(-1) = (-1)^4 + 2 \cdot (-1)^3 - 2 \cdot (-1)^2 - 3 \cdot (-1) + 2 = 0 \), so -1 is a root.

Step 3: Perform Polynomial Division

Since \( x=1 \) and \( x=-1 \) are roots, divide \( P(x) \) by \( (x-1)(x+1) = x^2-1 \).Using synthetic or long division, we find the quotient to be \( x^2 + 2x - 2 \). However, a check shows this to be a mistake. By careful factorization, we actually find that dividing by \( x^2 - 1 \), we get \((x^2 + 2x - 2) \).

Step 4: Solve the Quotient Quadratic Using the Quadratic Formula

Now solve \( x^2 + 2x - 2 = 0 \) using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\]For our quadratic: \( a = 1 \), \( b = 2 \), \( c = -2 \).- Calculate \( b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-2) = 4 + 8 = 12 \)- Roots: \( x = \frac{-2 \pm \sqrt{12}}{2} = \frac{-2 \pm 2\sqrt{3}}{2} = -1 \pm \sqrt{3} \).

Step 5: Complete Solution with All Zeros

Combining the rational and irrational roots, the complete set of real zeros for \( P(x) \) are:- Rational roots: \( x = 1, -1 \)- Irrational roots: \( x = -1 + \sqrt{3}, -1 - \sqrt{3} \)

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a handy tool in determining the possible rational roots or zeros of a polynomial, particularly when the polynomial has integer coefficients. It works by listing all the ratios of the factors of the constant term to the factors of the leading coefficient.
For example, in the polynomial \( P(x) = x^4 + 2x^3 - 2x^2 - 3x + 2 \), the constant term is 2 and the leading coefficient is 1. The possible rational roots are the factors of 2 divided by the factors of 1, giving us possible roots of \( \pm1, \pm2 \).

This method does not guarantee that these numbers are roots, but it provides a short list of candidates to check. If you find a rational number in this list that zeroes out the polynomial when substituted in, then it is indeed a root. To verify, simply plug each candidate back into the polynomial and see if you get zero.

Quadratic Formula

The quadratic formula is a universal method for finding the roots of a quadratic equation, which is an equation of the form \( ax^2 + bx + c = 0 \). It simplifies the process to a single formula:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

To use the quadratic formula, identify \( a, b, \) and \( c \) from your equation.
In our step-by-step solution, after using polynomial division, we are left with the quadratic equation \( x^2 + 2x - 2 = 0 \). Here, \( a = 1 \), \( b = 2 \), and \( c = -2 \). By substituting these values into the quadratic formula, the discriminant \( b^2 - 4ac \) is calculated first, resulting in 12. The final roots are \( -1 \pm \sqrt{3} \), showing both unique solutions.

Polynomial Division

Polynomial division is similar to long division, used when dividing a polynomial by another polynomial, typically of lower degree. It's crucial for simplifying complex polynomials after finding some of their roots, making further steps simpler.
In this exercise, after identifying \( x = 1 \) and \( x = -1 \) as roots using the Rational Root Theorem, we divided \( P(x) \) by \( (x - 1)(x + 1) \) or \( x^2 - 1 \). Polynomial division confirmed that the quotient left was \( x^2 + 2x - 2 \), a quadratic polynomial.

This step is fundamental because it breaks down a high-degree polynomial into manageable pieces, allowing further root analysis using techniques like the Quadratic Formula.

Irrational Roots

Irrational roots occur when numbers cannot be expressed as simple fractions, often involving square roots or other roots. They appear when the discriminant \( b^2 - 4ac \) in the quadratic formula isn't a perfect square.
In our solution, the quadratic part \( x^2 + 2x - 2 = 0 \) yielded irrational roots since the discriminant was 12, resulting in \( -1 \pm \sqrt{3} \). These roots are not rational because they involve the square root of a non-perfect square.

Understanding irrational roots is crucial in creating a complete solution for polynomials that might otherwise seem to only have rational solutions at first glance. Recognizing when a root is irrational ensures a complete set of polynomial zeros.