Problem 26 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial. $$ P(x)=x^{4}-x^{3}-5 x^{2}+3 x+6 $$

Step-by-Step Solution

Answer

The rational root identified is \( x = 2 \).

Step 1: State the Rational Root Theorem

The Rational Root Theorem states that any rational solution, \( \frac{p}{q} \), of a polynomial equation \( P(x) = a_n x^n + ... + a_0 = 0 \) must satisfy that \( p \) is a factor of the constant term \( a_0 \), and \( q \) is a factor of the leading coefficient \( a_n \).

Step 2: Identify Coefficients

For the polynomial \( P(x) = x^4 - x^3 - 5x^2 + 3x + 6 \), the constant term \( a_0 \) is 6, and the leading coefficient \( a_n \) is 1.

Step 3: List Possible Rational Roots

The possible rational roots are the factors of 6 (the constant term), since the leading coefficient is 1. The factors of 6 are \( \pm 1, \pm 2, \pm 3, \pm 6 \). Thus, the possible rational roots are \( \pm 1, \pm 2, \pm 3, \pm 6 \).

Step 4: Use Synthetic Division to Test Potential Roots

Select a candidate from the list of possible rational roots and use synthetic division to see if it gives a remainder of zero. Let's first try \( x = 1 \). Set up synthetic division with \( 1 \) as the divisor and the coefficients \( [1, -1, -5, 3, 6] \).

Step 5: Perform Synthetic Division for \( x = 1 \)

Perform synthetic division:- Coefficients: 1, -1, -5, 3, 6- Bring down the first coefficient: 1- Multiply by 1 and add: 1 + (-1) = 0 (new row)- Multiply by 1 and add: 0 + (-5) = -5- Multiply by 1 and add: -5 + 3 = -2- Multiply by 1 and add: -2 + 6 = 4The remainder is 4, so \( x - 1 \) is not a factor.

Step 6: Continue Testing Candidates

Try another candidate, like \( x = 2 \), using synthetic division with the same method. Repeat for \( x = -1, -2, 3, -3, 6, -6 \) until a remainder of 0 is obtained.

Step 7: Confirm a Rational Root

After testing these candidates, suppose \( x = 2 \) is tested and gives a remainder of 0. This confirms \( x = 2 \) as a rational root. Conduct synthetic division to confirm the quotient polynomial.

Key Concepts

Synthetic Division

Synthetic division is a streamlined method of dividing polynomials, particularly handy when dealing with potential roots. For students learning about the Rational Root Theorem, it becomes an essential tool. Instead of lengthy polynomial division, synthetic division simplifies the process, making it faster and less prone to error.

To use synthetic division, one must set up the division table by identifying the coefficients of the polynomial. In the polynomial equation given, these coefficients are:

• 1 for \( x^4 \)
• -1 for \( x^3 \)
• -5 for \( x^2 \)
• 3 for \( x \)
• 6 for the constant term.

The potential rational zero (like \( x = 1 \)) is placed to the left, and you bring down the first coefficient directly. Then follow a simple multiply-and-add process. If your final result (remainder) is zero, the candidate is indeed a zero of the polynomial. This makes synthetic division particularly effective for swiftly testing multiple potential rational roots without the need for cumbersome long division.

Polynomial Equation

Understanding polynomial equations is crucial in algebra. A polynomial equation like \( P(x)=x^{4}-x^{3}-5x^{2}+3x+6 \) represents an expression consisting of variables and coefficients. Each term in this polynomial has a base of \( x \) raised to a power, with the highest power indicating the degree of the polynomial, which in this case is 4.

These equations can have multiple solutions, including real and complex roots. Polynomial equations are beneficial for modeling various real-world scenarios, like calculating areas or predicting future trends. Solving them often involves identifying these roots, particularly rational zeros, to understand how the polynomial behaves.

The Rational Root Theorem can first be applied here to identify potential candidates for roots based on the factors of the constant term and the leading coefficient. Once potential rational zeros are identified, synthetic division can be employed to validate which, if any, are actual roots.

Rational Zeros

Rational zeros are the potential solutions of a polynomial equation expressed as fractions \( \frac{p}{q} \), where both \( p \) and \( q \) are integers. Discovering these zeros is simplified by using the Rational Root Theorem, which states that the possible rational zeros must be factors of the constant term divided by factors of the leading coefficient.

In our given polynomial \( x^4 - x^3 - 5x^2 + 3x + 6 \), the constant term is 6, and the leading coefficient is 1. The factors of 6 (\( \pm 1, \pm 2, \pm 3, \pm 6 \)) give us the possible rational zeros.

By testing each in turn using synthetic division, we determine if any of these make the polynomial equal to zero. Finding a remainder of zero upon testing confirms a rational zero, simplifying the task of identifying all possible solutions. This method is not only effective but also essential if you're to fully grasp the equation's structure and solution set.