Problem 72 - A Graphical Approach to College Algebra

Question

Use the given zero to completely factor \(P(x)\) into linear factors. $$\text { Zero: }-3 i ; P(x)=x^{5}+2 x^{4}+10 x^{3}+20 x^{2}+9 x+18$$

Step-by-Step Solution

Answer

The polynomial \(P(x)\) can be factored as \((x + 3i)(x - 3i)(x + 2)^2(x + 1)\).

Step 1: Use the given zero

Since complex roots come in conjugate pairs, the polynomial must also have the zero \(3i\). This gives us two factors: \((x - 3i)(x + 3i) = x^2 + 9\).

Step 2: Divide Polynomial by Factor

Using synthetic division or polynomial long division, divide \(P(x)\) by \(x^2 + 9\). This will reduce the degree of the polynomial by two.

Step 3: Complete Division Calculation

Carry out the division \(P(x) \div (x^2 + 9)\). You will find that \(P(x) = (x^2 + 9)(x^3 + 2x^2 + x + 2)\).

Step 4: Further Factorization

Factor the remaining cubic polynomial \(x^3 + 2x^2 + x + 2\). Use factoring by grouping or the Rational Root Theorem to attempt to find rational roots and factor completely.

Step 5: Verify Complete Factorization

The cubic piece can be factored as \((x + 2)^2(x + 1)\). Hence, \(P(x) = (x^2 + 9)(x + 2)^2(x + 1)\).

Step 6: Write Final Linear Factors

So the linear factors of \(P(x)\) are \((x + 3i)(x - 3i)(x + 2)(x + 2)(x + 1)\) or equivalently, \((x^2 + 9)(x + 2)^2(x + 1)\).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers and include quantities that have a real part and an imaginary part. The imaginary unit is denoted as \(i\), where \(i^2 = -1\). This means complex numbers can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers.

• The real part of the complex number is \(a\), and the imaginary part is \(bi\).
• Complex numbers are particularly valuable in solving polynomial equations that do not have real solutions.
• They are also crucial in electrical engineering, signal processing, and quantum mechanics.

In the given exercise, \(-3i\) is a zero of the polynomial. Because complex roots occur in conjugate pairs, \(3i\) is also a zero. The concept of conjugate pairs improves our ability to factor polynomials like \(P(x)\) into real-valued equations.

Polynomial Division

Polynomial division is a method used to divide a polynomial by another polynomial of equal or lower degree. It's analogous to the long division you might know from arithmetic, but there are some specific rules due to the variables involved.

• There are two common methods: synthetic division and long division.
• Synthetic division is often easier and faster, but it only works when dividing by linear terms like \(x - a\).
• Long division is more versatile and can be used with any polynomials.

In the exercise, we divided \(P(x)\) by \(x^2 + 9\) using these methods, which helps reduce the polynomial's degree and find its linear factors. This is crucial for breaking polynomials down into simpler components.

Rational Root Theorem

The Rational Root Theorem is a powerful tool for finding rational roots of a polynomial equation. It states that any rational solution \(\frac{p}{q}\) of a polynomial equation with integer coefficients is such that \(p\) (the numerator) is a factor of the constant term, and \(q\) (the denominator) is a factor of the leading coefficient.

• This theorem helps in finding potential rational roots to test by substituting into the polynomial.
• Once a rational root is found, it is used to factor the polynomial.
• This often reduces the degree of the polynomial, making it easier to solve or factor further.

By applying the Rational Root Theorem in the exercise, we attempted to find rational roots of the cubic polynomial \(x^3 + 2x^2 + x + 2\). This step is key to simplifying and factoring the polynomial completely.

Cubic Polynomials

Cubic polynomials are third-degree polynomials, which means they have the general form \(ax^3 + bx^2 + cx + d\). Solving and factoring cubic polynomials can be more challenging than linear or quadratic polynomials but can be simplified using specific techniques.

• Factoring cubic polynomials often involves recognizing patterns or using algebraic methods like synthetic division.
• For complete factorization, finding at least one real or rational root is necessary.
• Once a real root is discovered, we can perform polynomial division to factor the polynomial further.

In the exercise, after obtaining \(P(x)\) as \((x^2 + 9)(x^3 + 2x^2 + x + 2)\), the next task was to factor the cubic polynomial \(x^3 + 2x^2 + x + 2\). By finding suitable rational roots, we succeeded in simplifying it to \((x + 2)^2(x + 1)\), demonstrating the utility of such techniques.