Problem 47 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The real zeros are \(-2\), \(-1 + \sqrt{2}\), and \(-1 - \sqrt{2}\).

Step 1: Find the Rational Zeros

Using the Rational Root Theorem, first check which rational numbers could potentially be zeros of the polynomial \(P(x) = x^3 + 4x^2 + 3x - 2\). The possible rational zeros are the factors of the constant term \(-2\) divided by the factors of the leading coefficient \(1\). Thus, the possible rational zeros are \(\pm 1, \pm 2\).

Step 2: Test the Rational Zeros

Evaluate the polynomial at the possible rational zeros found in Step 1. Begin by testing \(x = 1\): \[P(1) = 1^3 + 4 \cdot 1^2 + 3 \cdot 1 - 2 = 6\]. This is not zero, so \(x = 1\) is not a zero.Next, test \(x = -1\): \[P(-1) = (-1)^3 + 4 \cdot (-1)^2 + 3 \cdot (-1) - 2 = -2\]. This is not zero.Next, test \(x = 2\): \[P(2) = 2^3 + 4 \cdot 2^2 + 3 \cdot 2 - 2 = 26\]. This is not zero.Now test \(x = -2\): \[P(-2) = (-2)^3 + 4 \cdot (-2)^2 + 3 \cdot (-2) - 2 = 0\]. This is zero, so one of the zeros is \(x = -2\).

Step 3: Perform Polynomial Division

Since \(x = -2\) is a root, divide \(P(x)\) by \(x + 2\) using synthetic division or polynomial division. Dividing gives the quotient of \(x^2 + 2x - 1\). So, \(P(x) = (x + 2)(x^2 + 2x - 1)\).

Step 4: Solve the Quadratic Equation

The next step is to find the zeros of the quadratic \(x^2 + 2x - 1\). To do this, use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=2\), and \(c=-1\). Calculate the discriminant: \(b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot (-1) = 8\). Substitute into the formula: \[x = \frac{-2 \pm \sqrt{8}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2}\].Thus, the zeros of the quadratic are \(-1 + \sqrt{2}\) and \(-1 - \sqrt{2}\).

Step 5: Summarize the Real Zeros

Combine the results of the previous steps: The real zeros of \(P(x) = x^3 + 4x^2 + 3x - 2\) are \(-2\), \(-1 + \sqrt{2}\), and \(-1 - \sqrt{2}\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a very helpful tool for finding possible rational zeros of a polynomial. It provides a list of potential candidates without having to try every possible number.
The theorem states that if a polynomial has a rational zero \( \frac{p}{q} \), where \( p \) and \( q \) are integers, then \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
For the polynomial \( P(x) = x^3 + 4x^2 + 3x - 2 \), the constant term is \(-2\) and the leading coefficient is \(1\). This means our potential rational zeros come from the factors of \(-2\) (which are \( \pm 1, \pm 2 \)) divided by the factors of \(1\) (simply \( \pm 1 \)). Thus, the possible rational zeros are \( \pm 1 \) and \( \pm 2 \).
After determining these candidates, the next step is testing each one to see which, if any, are actual zeros of the polynomial.

Polynomial Division

Once a rational zero of a polynomial is found, polynomial division becomes a crucial technique. It involves dividing the polynomial by a binomial to reduce its degree and make it easier to solve.
In our case, we discovered \(x = -2\) is a zero by substitution. This means \(x + 2\) is a factor of the polynomial \( P(x) = x^3 + 4x^2 + 3x - 2 \). We can use polynomial division here to divide \( P(x) \) by \( x + 2 \).
Completing this division results in a quotient of \( x^2 + 2x - 1 \). Now, the original polynomial can be written as \( P(x) = (x + 2)(x^2 + 2x - 1) \). By breaking it down, we have isolated a quadratic equation, simplifying the task of finding further roots.

Quadratic Formula

The quadratic formula is a straightforward method for finding zeros of a quadratic equation when it cannot be easily factored. The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation \(ax^2 + bx + c = 0\). For \( x^2 + 2x - 1 \), we set \(a = 1\), \(b = 2\), and \(c = -1\). Calculating the discriminant, \(b^2 - 4ac = 4 + 4 = 8\), which is positive, indicating two real and distinct solutions. Substituting these into the quadratic formula yields the roots \(-1 + \sqrt{2}\) and \(-1 - \sqrt{2}\). These solutions are the real zeros of the quadratic portion of the polynomial.

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a linear factor of the form \(x - c\). It is generally easier and quicker than traditional polynomial division, especially for higher degree polynomials.
In synthetic division, use the zero of \(x + 2\) (which is \(-2\)) to divide \(x^3 + 4x^2 + 3x - 2\). This method involves using the coefficients of the polynomial and systematically reducing them through a series of multiplications and additions.
By setting up the synthetic division, place the zero \(-2\) to the left and list the coefficients \(1, 4, 3, -2\). After completing the synthetic division, the result matches the more computationally intensive polynomial division: \(x^2 + 2x - 1\).
This confirms that \(x + 2\) is a factor, just like in the polynomial division, demonstrating the elegance and efficiency of synthetic division for solving polynomial equations.