Problem 5 - Precalculus

Question

An equation is given, followed by one or more roots of the equation. In each case, determine the remaining roots. $$x^{4}+10 x^{3}+38 x^{2}+66 x+45=0 ; x=-2+i$$

Step-by-Step Solution

Answer

The remaining roots are \( x = -2-i \) and \( x = -3\) (double root).

Step 1: Recognize the Given Root

The problem states that one root of the polynomial is \( x = -2+i \). Since the coefficients of the polynomial are all real, any complex roots must occur in conjugate pairs. Hence, \( x = -2-i \) is also a root of the polynomial.

Step 2: Construct the Quadratic Factor

Given the roots \( x = -2+i \) and \( x = -2-i \), we can form the quadratic factor of the polynomial using these roots. The factor is \((x - (-2+i))(x - (-2-i))\). Simplifying this using the difference of squares formula, we get:\[(x + 2 - i)(x + 2 + i) = (x+2)^2 - i^2 = x^2 + 4x + 4 + 1 = x^2 + 4x + 5.\]

Step 3: Perform Polynomial Division

Divide the original polynomial \( x^4 + 10x^3 + 38x^2 + 66x + 45 \) by the quadratic factor \( x^2 + 4x + 5 \) to find the remaining quadratic factor of the polynomial.

Step 4: Setup and Execute Long Division

Set up the long division of \( x^4 + 10x^3 + 38x^2 + 66x + 45 \) by \( x^2 + 4x + 5 \). Perform the division step by step:1. Divide \( x^4 \) by \( x^2 \) to get \( x^2 \).2. Multiply \( x^2 \) by \( x^2 + 4x + 5 \) to get \( x^4 + 4x^3 + 5x^2 \).3. Subtract this from the original polynomial, yielding \( 6x^3 + 33x^2 + 66x + 45 \).4. Divide \( 6x^3 \) by \( x^2 \) to get \( 6x \).5. Multiply \( 6x \) by \( x^2 + 4x + 5 \) to get \( 6x^3 + 24x^2 + 30x \).6. Subtract this from the result of step 3, yielding \( 9x^2 + 36x + 45 \).7. Divide \( 9x^2 \) by \( x^2 \) to get \( 9 \).8. Multiply \( 9 \) by \( x^2 + 4x + 5 \) to get \( 9x^2 + 36x + 45 \).9. Subtract to get a remainder of 0, confirming our division is correct.

Step 5: Identify the Remaining Roots

From the division, we get the quotient \( x^2 + 6x + 9 \). This can be factored as \((x+3)^2 \). Thus, the remaining roots are \( x = -3 \) (a repeated root).

Key Concepts

Complex Conjugates

When dealing with polynomials that have real coefficients, an important property is the occurrence of pairs of complex conjugate roots. A complex conjugate is formed by simply changing the sign of the imaginary part of a complex number.
In this exercise, we are provided with one complex root, \( x = -2+i \). Its conjugate partner will be \( x = -2-i \). These roots are not only solutions to the equation but also illustrate how complex solutions must balance within polynomials with real coefficients.
This is because when polynomial coefficients are real, any imaginary part introduced by a root must be countered by another root with an opposite imaginary part, ensuring that the overall polynomial remains real.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear factor, though it is not directly applicable for division with a quadratic factor like in this exercise. It’s ideal for when you need to divide by expressions of the form \( x - c \).
While synthetic division offers a streamlined approach, it’s more relevant for dividing polynomials when we have a simple linear root and need to check for further roots or factors quickly.
Despite not being used directly here, having a grounding in synthetic division is valuable as it builds intuition about polynomial division and roots.

Quadratic Factor

Identifying the quadratic factor based on complex roots is crucial. With complex roots \( x = -2+i \) and \( x = -2-i \), a quadratic can be constructed. By using these roots, we derive the quadratic factor as:

• The expression \((x + 2 - i)(x + 2 + i)\) applies the difference of squares.
• Thus, \((x+2)^2 - (i)^2\) simplifies to \( x^2 + 4x + 5 \).

This quadratic factor reveals more about the polynomial's structure, simplifying it for further division and analysis. Understanding how to construct such factors gives tools to unravel more complex polynomials.

Long Division

Long division of polynomials is an algebraic method used to divide a higher-degree polynomial by a lower-degree polynomial, similar to numerical long division.
The process is sequential and iterative:

• Begin by dividing the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by the result and subtract it from the dividend.
• Repeat the process with the new, smaller polynomial.

In the exercise, this method is employed to divide \( x^4 + 10x^3 + 38x^2 + 66x + 45 \) by the quadratic factor \( x^2 + 4x + 5 \).
This procedure helps simplify the original polynomial, allowing one to find the remaining factor \( x^2 + 6x + 9 \) and hence determine the remaining roots. Understanding long division is essential for exploring how factors divide into larger polynomials.