Problem 14 - A Graphical Approach to Precalculus with Limits

Question

One or more zeros are given for each polynomial. Find all remaining zeros. \(P(x)=x^{3}-5 x^{2}+17 x-13 ; \quad 1\) is a zero.

Step-by-Step Solution

Answer

The remaining zeros of the polynomial are \(2+3i\) and \(2-3i\).

Step 1: Understanding the Problem

We need to find all zeros of the polynomial \(P(x) = x^3 - 5x^2 + 17x - 13\) given that \(1\) is one of the zeros. This means if we substitute \(x = 1\) into \(P(x)\), it should equal zero.

Step 2: Perform Synthetic Division

Since \(1\) is a zero of the polynomial, divide \(P(x)\) by \(x-1\) using synthetic division to simplify \(P(x)\). Use the coefficients \((1, -5, 17, -13)\) in the synthetic division with \(1\) as the divisor. This division will provide a quotient which is a quadratic polynomial.

Step 3: Check Result of Synthetic Division

The division process results in the linear and constant terms which will be used in the next step. After synthetic division, the quotient is \(x^2 - 4x + 13\) and the remainder is zero, confirming \(1\) is indeed a zero.

Step 4: Solve the Quotient using the Quadratic Formula

The remaining quotient is \(x^2 - 4x + 13 = 0\). Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), substitute \(a = 1\), \(b = -4\), and \(c = 13\). This solves to \(x = \frac{4 \pm \sqrt{-36}}{2}\).

Step 5: Simplify Complex Roots

The discriminant \(-36\) is negative, indicating the roots are complex. Simplify to find the complex roots: \(x = 2 \pm 3i\). So the remaining zeros are \(2+3i\) and \(2-3i\).

Key Concepts

Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly useful when dividing by a linear factor like \( x - c \). It streamlines the process and reduces laborious calculations.
Here's how it works:

• Write down only the coefficients of the polynomial's terms. For example, for the given polynomial \( P(x) = x^3 - 5x^2 + 17x - 13 \), use the coefficients (1, -5, 17, -13).
• The divisor in synthetic division is the zero that you know. Here, it's 1, corresponding to the factor \( x-1 \).
• Place this zero out to the left. Your task is to determine the remaining coefficients of the polynomial when divided by this linear factor.
• Bring the leading coefficient (1 in this case) straight down.
• Multiply it by the divisor (1), and add this value to the next coefficient (-5) in the polynomial.
• Continue this process across the row, using each new value to multiply by the divisor and adding to the next coefficient.

If done correctly, the last number should be zero, indicating a perfectly even division with no remainder. Synthetic division here transforms \( P(x) \) into the quotient \( x^2 - 4x + 13 \). This step lays the groundwork for identifying remaining zeros.

Quadratic Formula

To find the zeros of a quadratic equation, such as the quotient polynomial \( x^2 - 4x + 13 \), you'll employ the quadratic formula. This robust formula is essential for solving second-degree polynomials:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• Here, \( a, b, \) and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). For \( x^2 - 4x + 13 \), we have \( a = 1 \), \( b = -4 \), and \( c = 13 \).
• Substitute these values into the equation to determine the values of \( x \).
• Calculate the discriminant, \( b^2 - 4ac \). For our equation, it's \(-36\), signaling that this quadratic has complex roots.

The quadratic formula allows you to handle any quadratic equation, including those whose solutions involve complex numbers, as seen in this exercise.

Complex Roots

Complex roots come into play when solving equations with a negative discriminant (\( b^2 - 4ac < 0 \)). They involve imaginary numbers, represented by \( i = \sqrt{-1} \).
Here's how you identify and work with complex roots:

• When using the quadratic formula, a negative discriminant leads to a square root of a negative number, such as \( \sqrt{-36} \), which can be rewritten as \( 6i \).
• For the quadratic \( x^2 - 4x + 13 \), the solutions resulting from the quadratic formula are \( x = 2 \pm 3i \), where you split the square root into its imaginary component.
• These complex numbers show up as pairs: \( 2 + 3i \) and \( 2 - 3i \). They are the remaining zeros of the polynomial \( P(x) \).
• Understanding complex numbers entails grasping the concept of combining real and imaginary parts, crucial in various fields of science and engineering.

Complex roots emphasize the polynomial’s depth, illustrating that even polynomials with real coefficients can have solutions residing in the complex plane.