Problem 65 - A Graphical Approach to Precalculus with Limits

Question

For each polynomial at least one zero is given. Find all others analytically. $$P(x)=3 x^{3}+5 x^{2}-3 x-2 ;-2$$

Step-by-Step Solution

Answer

The zeros are \(-2\), \(\frac{1 + \sqrt{13}}{6}\), and \(\frac{1 - \sqrt{13}}{6}\).

Step 1: Understand the Problem

We need to find all the zeros of the polynomial \(P(x) = 3x^3 + 5x^2 - 3x - 2\). We know that \(-2\) is a zero of the polynomial, meaning \(P(-2) = 0\).

Step 2: Perform Polynomial Long Division

Since \(-2\) is a zero, \((x + 2)\) is a factor of the polynomial. Perform polynomial long division of \(P(x)\) by \((x + 2)\) to find the quotient polynomial \(Q(x)\).

Step 3: Execute Division

Divide \(3x^3 + 5x^2 - 3x - 2\) by \(x + 2\). 1. Divide \(3x^3\) by \(x\) to get \(3x^2\).2. Multiply \(3x^2\) by \(x + 2\) to get \(3x^3 + 6x^2\).3. Subtract \(3x^3 + 6x^2\) from \(3x^3 + 5x^2\) to get \(-x^2\).4. Bring down \(-3x\) and repeat: divide \(-x^2\) by \(x\) to get \(-x\).5. Multiply \(-x\) by \(x + 2\) to get \(-x^2 - 2x\).6. Subtract to get \(-x\) and bring down \(-2\).7. Divide \(-x\) by \(x\) to get \(-1\); multiply to get \(-x - 2\). The remainder is 0.The quotient is \(3x^2 - x - 1\).

Step 4: Solve the Quadratic Equation

We obtained \(3x^2 - x - 1\) from the division. We need to solve this quadratic equation to find the remaining zeros of \(P(x)\). Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a = 3\), \(b = -1\), and \(c = -1\).

Step 5: Calculate the Discriminant

Calculate the discriminant \(\Delta = b^2 - 4ac\) for the quadratic equation. \[ \Delta = (-1)^2 - 4 \cdot 3 \cdot (-1) = 1 + 12 = 13 \]

Step 6: Find the Roots Using the Quadratic Formula

Use the quadratic formula with \(\Delta = 13\). \[ x = \frac{1 \pm \sqrt{13}}{6} \]This gives the two roots: \(x = \frac{1 + \sqrt{13}}{6}\) and \(x = \frac{1 - \sqrt{13}}{6}\).

Step 7: Compile All Zeros

The polynomial \(P(x) = 3x^3 + 5x^2 - 3x - 2\) has zeros at \(-2\), \(\frac{1 + \sqrt{13}}{6}\), and \(\frac{1 - \sqrt{13}}{6}\).

Key Concepts

Polynomial Long Division

Polynomial long division is a valuable method for dividing one polynomial by another, especially when one root is known. In the given problem, we have a polynomial \( P(x) = 3x^3 + 5x^2 - 3x - 2 \) and we know \(-2\) is a zero. This means \((x + 2)\) is a factor of the polynomial. The division process helps us simplify the polynomial, making it easier to find the other zeros.

To perform polynomial long division:

• Start by dividing the leading term of the dividend \(3x^3\) by the leading term of the divisor \(x\) to get \(3x^2\).
• Multiply \(3x^2\) by \(x + 2\) and subtract the result from the original polynomial.
• Continue the process with the new polynomial formed after subtraction until reaching a remainder of zero or the degree of the remainder is less than the divisor.

After completing the division, the quotient \(3x^2 - x - 1\) provides a simpler polynomial to solve, which can be addressed using other techniques, such as the quadratic formula.

Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form \(ax^2 + bx + c = 0\). When polynomial long division reveals a quadratic equation like \(3x^2 - x - 1\), this formula becomes essential. It is valuable because it can find zeros even in cases where factoring is challenging.

The formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation. In this example, \(a = 3\), \(b = -1\), and \(c = -1\). By calculating these values, the formula will yield the possible zeros.

Understanding how to apply the quadratic formula effectively allows for the determination of roots for any quadratic equation confidently, no matter the complexity of the numbers involved.

Discriminant Calculation

The discriminant is a key component of the quadratic formula, represented as \(\Delta = b^2 - 4ac\). It helps determine the nature and number of roots of a quadratic equation. Before using the quadratic formula, calculating the discriminant provides insight into what to expect:

• If \(\Delta > 0\): The quadratic equation has two distinct real roots.
• If \(\Delta = 0\): There is exactly one real root (a repeated root).
• If \(\Delta < 0\): No real roots exist; the solutions are complex.

In the problem, \(\Delta = 13\) indicates two distinct real roots. This clarity simplifies the process by ensuring that the resulting roots from the quadratic formula will be real and distinct, allowing the polynomial's complete set of zeros to be accurately determined.