Problem 50 - Algebra and Trigonometry

Question

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{4}-2 x^{3}-2 x^{2}-2 x-3 $$

Step-by-Step Solution

Answer

The zeros are found using the Rational Root Theorem and synthetic division.

Step 1: Use the Rational Root Theorem

The Rational Root Theorem suggests that any rational root of the polynomial \(P(x)\) is a factor of the constant term, \(-3\), divided by a factor of the leading coefficient, \(1\). Hence, the possible rational roots are \(\pm 1, \pm 3\).

Step 2: Test Possible Rational Roots

Evaluate the polynomial at each possible root from the previous step. For example, evaluate \(P(1)\), \(P(-1)\), \(P(3)\), and \(P(-3)\). Calculating, \(P(1) = 1^4 - 2\times1^3 - 2\times1^2 - 2\times1 - 3 eq 0\), indicating \(x=1\) is not a root. Similarly test the other values.

Step 3: Use Synthetic Division

After finding a root, use synthetic division to factor the polynomial. Suppose re-evaluating showed \(P(-1) = 0\), this indicates \(x + 1\) is a factor. Use synthetic division to divide \(P(x)\) by \(x + 1\) to get another polynomial of degree 3.

Step 4: Factor the Resultant Polynomial

Divide the original polynomial by \(x + 1\) using synthetic division and factor the resultant polynomial. For example, check if further factorization is possible by checking if it factorizes into quadratic form or through other roots.

Step 5: Solve the Quadratic Polynomial

Solve the resultant quadratic polynomial either by factoring, completing the square, or using the quadratic formula. For example, if it simplifies to \(x^2 + x + 1 = 0\), the quadratic formula can be applied: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a useful tool in finding the zeros of a polynomial, particularly when dealing with integer coefficients. It helps to identify potential rational solutions by analyzing the possible fractions that could be roots.

• The theorem states that any rational solution \( \frac{p}{q} \) of the polynomial equation \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_0 = 0 \) (where coefficients are integers) has a numerator \( p \) which is a factor of the constant term \( a_0 \).
• The denominator \( q \) is a factor of the leading coefficient \( a_n \).

In our original problem, the polynomial is \( P(x)=x^{4}-2x^{3}-2x^{2}-2x-3 \). Here, the constant term is \,\(-3\), and the leading coefficient is \,\(1\). Thus, the possible rational roots are gathered by taking factors of \,\(-3\): \,\(\pm 1, \pm 3\). These potential roots are simplicity themselves – they are whole numbers! To proceed, you would evaluate these candidates by substituting them into the polynomial to see if they yield zero. This method helps find any rational roots without exhaustive trial and error.

synthetic division

Synthetic division is a simplified form of polynomial division, particularly useful when dividing by a linear factor of the form \( x - c \). It minimizes the complexity of regular long division by working specifically with the coefficients. Here's how it works:

• Begin with the root you found using the Rational Root Theorem; say, \( x = c \).
• Write down ONLY the coefficients of the polynomial.
• Set up a row to perform arithmetic under the coefficients, dropping the first one immediately.
• Multiply \( c \) by the previous result to the added value above the row, and place it under the next coefficient.
• Continue this step-by-step process across, adding downward each time, until new coefficients for the quotient polynomial are reached.

For example, if \( x = -1 \) is confirmed as a root, we divide \( P(x) \) by \( x + 1 \) using these steps. If done correctly, the remainder should be 0, and the polynomial reduces. This reveals another straightforward polynomial of lower degree to further analyze or solve.

quadratic formula

The Quadratic Formula is a fundamental tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It provides a clear method to find solutions by substituting the coefficients into a simple formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Using the formula involves a few essential steps:

• Identify \( a \), \( b \), and \( c \) from the quadratic equation.
• Calculate the discriminant, \( b^2 - 4ac \).
• If the discriminant is positive, expect two real and distinct solutions.
• If it equals zero, expect one real solution (also known as a repeated root).
• If negative, expect complex solutions.

In context with our exercise, after performing synthetic division, we could arrive at a quadratic polynomial. For instance, consider processing something like \( x^2 + x + 1 = 0 \). Here, inserting \( a = 1 \), \( b = 1 \), and \( c = 1 \) into the formula results in solutions. This formula aids in solving such reduced quadratic forms quickly once larger polynomials are simplified.