Problem 46 - Algebra and Trigonometry

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example 3(a). $$ P(x)=x^{5}-4 x^{4}-x^{3}+10 x^{2}+2 x-4 $$

Step-by-Step Solution

Answer

The real zeros of the polynomial are 2 and 1.

Step 1: Identify Polynomial Roots Potentially Using Factoring

The given polynomial is \(P(x) = x^5 - 4x^4 - x^3 + 10x^2 + 2x - 4\). Before using the quadratic formula, we attempt to factor out any common elements or find obvious roots, possibly through testing simple values by substitution.

Step 2: Use the Rational Root Theorem

The Rational Root Theorem helps us identify potential rational roots by considering the factors of the constant term \(-4\) and the leading coefficient, which is \(1\). Test \(\pm1, \pm2, \pm4\) by substitution, and find that \(x = 2\) is a root.

Step 3: Divide Polynomial by \(x - 2\)

Since \(x = 2\) is a root, divide the polynomial \(P(x)\) by \(x - 2\) to reduce its degree. Use synthetic or long division to perform the division, resulting in a quotient of \(x^4 - 2x^3 - 5x^2 - 4\).

Step 4: Factor Remaining Polynomial

Next, factor the quadratic expression resulting from the division: \(x^4 - 2x^3 - 5x^2 - 4\). Further factorization or solving may reveal additional roots.

Step 5: Use Quadratic Formula if Needed

If the polynomial cannot be easily factored further, apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to any quadratic factor to find the remaining roots. For \(x^2 + x - 2\), solve this to find any remaining real zeros.

Key Concepts

Quadratic Formula

The quadratic formula is a tool used to find the roots or zeros of quadratic equations. These are polynomials of degree 2, typically expressed as \(ax^2 + bx + c = 0\). The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula utilizes the coefficients \(a\), \(b\), and \(c\) from the quadratic polynomial. The term under the square root, called the discriminant \((b^2 - 4ac)\), indicates the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root.
• If it is negative, the roots are complex and not real.

For the exercise polynomial, once simplified through initial factorization and division, any remaining quadratic expressions can be solved for their roots using this formula. It's especially handy when direct factorization becomes cumbersome.

Rational Root Theorem

The Rational Root Theorem is a powerful method to predict potential rational roots of a polynomial. For a polynomial of the form \(a_nx^n + a_{n-1}x^{n-1} + \dots + a_0\), it states that any rational solution, expressed as \(\frac{p}{q}\), requires \(p\) to be a factor of the constant term \(a_0\), and \(q\) to be a factor of the leading coefficient \(a_n\).
In our provided polynomial, \(P(x) = x^5 - 4x^4 - x^3 + 10x^2 + 2x - 4\), the constant term \(-4\) and the leading coefficient, which is \(1\), guide the potential rational roots. Testing these values is straightforward: simply substitute each possible rational root into the polynomial and check if it results in zero. This simple process significantly narrows down the possibilities and helps in determining at least one root, as was found with \(x = 2\). This discovery then facilitates polynomial division to further simplify the equation.

Polynomial Division

Polynomial division is similar to long division but is used for dividing polynomials such as those we encounter during the problem-solving process. The goal is to divide a given high-degree polynomial by a simpler polynomial, usually of degree 1 (a linear polynomial). This helps in breaking down the polynomial into factors.
In our problem scenario, we use polynomial division after finding \(x = 2\) as a root using the Rational Root Theorem. We divide the original polynomial, \(P(x) = x^5 - 4x^4 - x^3 + 10x^2 + 2x - 4\), by \(x - 2\), resulting in a reduced polynomial \(x^4 - 2x^3 - 5x^2 - 4\).
This division can be accomplished through long division, akin to the arithmetic long division process, or by using synthetic division, which is more efficient for linear divisors. Each step systematically reduces the degree of the polynomial, making further factorization or application of the quadratic formula feasible.

Synthetic Division

Synthetic division is a streamlined technique for dividing a polynomial by a linear divisor of the form \(x - c\). It simplifies the work involved, especially when compared to long division, and is perfectly suited for problems where the Rational Root Theorem identifies a potential root.
The method involves:

• Writing down the coefficients of the polynomial.
• Using the identified root \(c\) to simplify the division process.
• Combining and eliminating coefficients step by step according to the synthetic division rules.

For our exercise, once \(x = 2\) was recognized as a root, synthetic division efficiently confirmed the remaining polynomial as \(x^4 - 2x^3 - 5x^2 - 4\). Each calculation aligns to give the result quickly without the need for repetitive operations seen in long division.