Problem 22 - Precalculus Mathematics for Calculus

Question

Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. $$P(x)=x^{3}-x^{2}+x$$

Step-by-Step Solution

Answer

Zero: \(x = 0\) with multiplicity 1. The quadratic has complex zeros.

Step 1: Identify the Greatest Common Factor (GCF)

First, notice that each term in the polynomial \(P(x) = x^3 - x^2 + x\) contains the variable \(x\). Thus, the greatest common factor (GCF) is \(x\). Factor \(x\) out of the polynomial.

Step 2: Factor out the GCF

Factor the GCF, \(x\), from the polynomial: \[ P(x) = x(x^2 - x + 1) \]. This simplifies the polynomial and leaves a quadratic expression inside the parentheses.

Step 3: Check the Quadratic for Factorability

The quadratic \(x^2 - x + 1\) needs to be checked for factorability. Calculate the discriminant \(b^2 - 4ac\). For this quadratic, \(a = 1\), \(b = -1\), \(c = 1\), so the discriminant is \((-1)^2 - 4(1)(1) = 1 - 4 = -3\). A negative discriminant implies it cannot be factored over the real numbers.

Step 4: Identify the Zeros from the Factorization

Since the quadratic \(x^2 - x + 1\) cannot be factored further over the real numbers, look at the linear factor that we factored out, \(x\). This gives us an immediate zero: \(x = 0\).

Step 5: Determine the Multiplicity of Each Zero

The factor \(x\) implies a zero at \(x = 0\) with multiplicity 1, since it appears once in the linear factorization with no repeated powers.

Key Concepts

Greatest Common Factor

The Greatest Common Factor, often known as the GCF, is a key concept in simplifying polynomials. The GCF is the largest factor shared by all terms of a polynomial. In simpler terms, it's the biggest expression that divides every term of the polynomial without leaving a remainder. Finding the GCF is usually the first step toward factoring a polynomial, as it helps to simplify the expression by "pulling out" the commonalities among the terms.

For example, in the given polynomial \(P(x) = x^3 - x^2 + x\), remove the common factor which is \(x\). This process extracts the GCF \(x\) from each term:

• From \(x^3\), we're left with \(x^2\).
• From \(-x^2\), we're left with \(-x\).
• From \(x\), we're left with \(1\).

Thus, we can factor the polynomial into \(x(x^2 - x + 1)\), simplifying the polynomial while maintaining equivalency.

Discriminant in Quadratics

The discriminant is a crucial value when dealing with quadratic expressions to determine whether they can be factored into real numbers. It's calculated using the formula \(b^2 - 4ac\) for a quadratic equation in the standard form \(ax^2 + bx + c\). This calculation tells us about the nature of the roots without actually solving the equation.

• If the discriminant is positive, the quadratic has two distinct real roots and is factorable over the real numbers.
• If the discriminant is zero, the quadratic has exactly one real root, implying it’s a perfect square.
• If the discriminant is negative, as in this exercise, it means no real roots exist, and the quadratic cannot be factored over the real numbers.

In the polynomial \(x^2 - x + 1\), substituting \(a = 1\), \(b = -1\), and \(c = 1\), we find the discriminant is \(-3\). This indicates the quadratic cannot be further factored into real-number linear factors.

Multiplicity of Zeros

Understanding the multiplicity of zeros is important for fully characterizing the roots of a polynomial function. A zero's multiplicity refers to how many times that particular zero appears in the factored form of the polynomial.

Multiplicity impacts the graph of the polynomial:

• A zero with odd multiplicity crosses the x-axis.
• A zero with even multiplicity merely touches the x-axis.

In the polynomial \(P(x) = x(x^2 - x + 1)\), which we factored from \(x^3 - x^2 + x\), the zero at \(x = 0\) appears directly from the \(x\) factor, and no other powers of \(x\) show up in the factorization. Thus, this zero has a multiplicity of 1, meaning it crosses the x-axis exactly once at \(x = 0\). Understanding multiplicities is crucial when sketching polynomial functions, as they determine the behavior of the graph at x-axis intersections.