Problem 75 - A Graphical Approach to College Algebra

Question

Use the given zero to completely factor \(P(x)\) into linear factors. $$\text { Zero: } 1+i ; \quad P(x)=x^{4}-2 x^{3}+3 x^{2}-2 x+2$$

Step-by-Step Solution

Answer

The linear factors of \(P(x)\) are \((x - (1+i))(x - (1-i))(x - i)(x + i)\).

Step 1: Recognize Conjugate

Since the problem provides a complex zero, \(1+i\), and \(P(x)\) has real coefficients, the conjugate, \(1-i\), is also a zero.

Step 2: Form Quadratic Factor

Using the complex roots, we form the quadratic factor: \((x - (1+i))(x - (1-i))\). Simplifying this gives: \((x - 1 + i)(x - 1 - i) = (x - 1)^2 - i^2 = (x - 1)^2 + 1 = x^2 - 2x + 2\).

Step 3: Polynomial Division

To factor \(P(x)\) completely, divide \(P(x)\) by \(x^2 - 2x + 2\) using polynomial long division.

Step 4: Perform Division

Divide \(x^4 - 2x^3 + 3x^2 - 2x + 2\) by \(x^2 - 2x + 2\):1. Divide the leading term of the dividend by the leading term of the divisor: \(x^4/x^2 = x^2\).2. Multiply \(x^2 \) by \(x^2 - 2x + 2\) and subtract from the dividend.3. Repeat the process with the new polynomial after subtraction.4. Continue until the remainder has lower degree than \(x^2 - 2x + 2\).

Step 5: Complete the Factorization

The result of the division should be \(x^2 + 1\), indicating that \(P(x) = (x^2 - 2x + 2)(x^2 + 1)\). Factor \(x^2 + 1\) further into \((x - i)(x + i)\).

Step 6: Write the Polynomial in Linear Factors

Combine all the factors to express \(P(x)\) completely in linear factors: \(P(x) = (x - (1+i))(x - (1-i))(x - i)(x + i)\).

Key Concepts

Complex Conjugates

When dealing with polynomials that have real coefficients and complex zeros, the concept of complex conjugates plays a crucial role. A complex conjugate of a number is formed by changing the sign of the imaginary part. For example, the complex conjugate of the zero \(1+i\) is \(1-i\). These conjugate pairs are always present when we have real polynomial coefficients.

Key things to remember about complex conjugates in this context:

• For every complex zero \(a+bi\), there is a conjugate \(a-bi\).
• The presence of a complex zero in a real polynomial implies its conjugate must also be a zero.

Using complex conjugates helps to simplify polynomial expressions and assist in factorization.

Polynomial Division

Polynomial division is a method used to divide polynomials much like long division with numbers. Here, it is used to simplify the polynomial by dividing it by another polynomial. This step is crucial to determine other factors that simplify the complete polynomial.

To perform polynomial division, follow these steps:

• Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
• Multiply the entire divisor by this term and subtract from the original dividend.
• Repeat the process with the new remainder, updating each time until you can't divide further.

This division helps in breaking down the polynomial into simpler components that can then be factored further.

Linear Factors

Linear factors are expressions of the form \(x - c\), where \(c\) is a root of the polynomial. Converting a polynomial fully into linear factors is a way of expressing its roots directly.

To express any polynomial as a product of its linear factors, follow these steps:

• Identify all the zeros (roots) of the polynomial.
• For each zero, express it as a linear factor \(x-c\).
• The product of all these linear factors will give the factored form of the polynomial.

In the exercise provided, after identifying all possible roots, you can express the polynomial as a product of these linear factors, thereby making it easier to understand and apply.

Quadratic Factor

A quadratic factor is a polynomial of degree 2 of the form \(ax^2 + bx + c\). Sometimes, when factoring polynomials, an expression naturally reduces to a quadratic form, especially when complex conjugates are involved.

Consider these steps to form a quadratic factor:

• Using the complex conjugates that are zeros of the polynomial, form the expression \((x - a - bi)(x - a + bi)\).
• Expand and simplify this expression to get a quadratic factor.

In many cases, this helps simplify and reduce polynomials to a form that can be further divided using polynomial division.

Polynomial Roots

Polynomial roots, also known as zeros, are the values of \(x\) that make the polynomial equal to zero. Identifying these roots is critical to factorizing polynomials fully.

Here's how you determine and use polynomial roots:

• Set your polynomial equation \(P(x) = 0\).
• Solve for the values of \(x\) that satisfy this equation.
• Each solution represents a root of the polynomial.

This exercise highlights finding complex roots and their usage in expressing the polynomial in terms of linear factors. Understanding roots not only helps in solving equations but also enhances comprehension of the behavior of polynomial functions.