Problem 63 - Algebra and Trigonometry

Question

A polynomial P is given. (a) Factor P into linear and irreducible quadratic factors with real coefficients. (b) Factor P completely into linear factors with complex coefficients. \(P(x)=x^{3}-5 x^{2}+4 x-20\)

Step-by-Step Solution

Answer

(a) Real factors: \((x - 5)(x^2 + 4)\); (b) Complex factors: \((x - 5)(x - 2i)(x + 2i)\).

Step 1: Find the Roots

To factor the polynomial \( P(x) = x^3 - 5x^2 + 4x - 20 \), we first find its roots. Use the Rational Root Theorem to test possible rational roots, which are the factors of the constant term, -20 (\( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \)). Test these values until you find a root.

Step 2: Verify a Root

After testing, \( x = 5 \) is a root of the polynomial. This can be verified by substituting \( x = 5 \) into \( P(x) \) and checking if it yields zero: \( 5^3 - 5 \times 5^2 + 4 \times 5 - 20 = 0 \).

Step 3: Polynomial Division

Since \( x = 5 \) is a root, divide \( P(x) \) by \( (x - 5) \) to find the quotient polynomial. Performing the division gives \( x^2 + 4 \).

Step 4: Factorization into Linear and Quadratic

The quotient from the division, \( x^2 + 4 \), is irreducible over the reals, meaning it can't be factored further into real linear factors. Therefore, the factorization of \( P(x) \) with real coefficients is \((x - 5)(x^2 + 4)\).

Step 5: Factorization into Linear Complex Factors

To factor completely over the complex numbers, note that \( x^2 + 4 \) can be rewritten using complex numbers as \((x - 2i)(x + 2i)\). Thus, the polynomial fully factored over the complex numbers is \((x - 5)(x - 2i)(x + 2i)\).

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a valuable tool for solving polynomial equations. Simply put, it offers a potential list of rational solutions, or roots, for a given polynomial. These potential roots are the ratios of integer factors of the constant term to integer factors of the first leading coefficient.

For example, consider the polynomial \( P(x) = x^3 - 5x^2 + 4x - 20 \). The constant term here is -20, and the leading coefficient is 1. According to this theorem, any rational solution of \( P(x) \) is a fraction made from factors of -20 divided by factors of 1.

Therefore, possible rational roots are: \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \). It simplifies the process significantly, narrowing the potential roots down, and making it easier to find actual ones by testing these candidates.

Real Coefficients

Real coefficients in a polynomial mean that all the numbers involved are real numbers. Real numbers include both finite decimals, fractions, integers, and irrational numbers such as \( \sqrt{2} \).

When factoring polynomials with real coefficients, try to find representations that do not involve imaginary numbers. In the case of the polynomial \( P(x) = x^3 - 5x^2 + 4x - 20 \), we found its factorization with real coefficients as \((x - 5)(x^2 + 4)\).

Here, \( x^2 + 4 \) is an irreducible quadratic over the real numbers because it cannot be factored further without using imaginary numbers. An irreducible quadratic has no real roots and does not break down further except into a format involving \( i \), the imaginary unit.

Complex Coefficients

Complex coefficients involve using complex numbers, which include the imaginary unit \( i \). The imaginary unit is defined by \( i^2 = -1 \).

Polynomials can be factored completely into linear factors using complex coefficients. For our polynomial \( P(x) = x^3 - 5x^2 + 4x - 20 \), we found its complete factorization as \((x - 5)(x - 2i)(x + 2i)\).

Here, \( x^2 + 4 \) splits into \((x - 2i)(x + 2i)\), incorporating imaginary numbers. Using complex coefficients allows any polynomial to be broken down into only linear factors, all involving real or complex numbers.

Irreducible Quadratic

An irreducible quadratic is a quadratic polynomial that cannot be factored further using real numbers. These quadratics do not have real roots; their discriminant, in the form \( b^2 - 4ac \), is negative, indicating complex solutions.

For our polynomial example, the quadratic \( x^2 + 4 \) is irreducible over the real numbers because it does not factor into real-number linear terms. Remember, when you have a quadratic like \( x^2 + c \) with \( c > 0 \), it is generally irreducible unless confronted with non-real roots.

When working solely within the realm of real numbers, such quadratics remain in their offered form as part of the polynomial's factorization.

Polynomial Division

Polynomial division is similar to long division but uses polynomials instead of numbers. It is used to divide one polynomial by another, often to simplify or factor the polynomial.

In the context of our problem, we used polynomial division to divide \( P(x) = x^3 - 5x^2 + 4x - 20 \) by \( (x - 5) \), obtaining the quotient \( x^2 + 4 \). This process helps isolate factored components and reveals hidden polynomials within the original expression.

The key is to systematically divide, just as in ordinary division, to align powers and subtract accordingly. Through this method, you can discover hidden factors and help further simplify or solve polynomial equations.