Problem 43 - Precalculus : Building Concepts and Connections

Question

Let \(x-\frac{1}{2}\) be a factor of a polynomial function \(p(x) .\) Find \(p\left(\frac{1}{2}\right).\)

Step-by-Step Solution

Answer

Hence if \(x-\frac{1}{2}\) is a factor of \(p(x)\) then the value of \(p\left(\frac{1}{2}\right)\) would be \(0\).

Step 1: Identify the Given Information

This problem refers to the Factor Theorem, which relates the zeros of a polynomial to its factors. We have been given that \(x-\frac{1}{2}\) is a factor of a polynomial function \(p(x)\).

Step 2: Apply the Factor Theorem

The Factor Theorem states that a polynomial function \(p(x)\) has a factor \(x-c\) if and only if \(p(c)=0\). In this problem, it is given that \(x-\frac{1}{2}\) is a factor of \(p(x)\). So, according to the Factor Theorem, the value \(p(c)\) will be \(0\) where \(c = \frac{1}{2}\). That is, \(p\left(\frac{1}{2}\right)=0\).

Key Concepts

Polynomial Function

A polynomial function is an expression that involves variables and coefficients, connecting them using mathematical operations such as addition, subtraction, and multiplication. These functions play a fundamental role in algebra and more advanced mathematics. A polynomial function can take the general form

• \( p(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \)

where \( a_n, a_{n-1}, \ldots, a_0 \) are coefficients, and \( n \) is a non-negative integer representing the highest power of the variable \( x \). The highest power, or the degree, determines the most significant characteristic of a polynomial, dictating its graph's shape and behavior as \( x \) becomes very large or very small.
Polynomials are incredibly versatile. They help in modeling real-world situations where a quantity changes following a smooth curve. Every polynomial function has at least one root or zero, if it is not a constant, and can be completely factored into linear terms, when possible, over the set of complex numbers.

Zeros of a Polynomial

The zeros of a polynomial are the values of \( x \) for which the function evaluates to zero. They are also known as roots and are crucial for understanding the properties of polynomial functions.
To find a polynomial's zeros, you can use several methods like factoring, synthetic division, or the quadratic formula (for quadratics). The Factor Theorem, especially, is an essential tool in identifying these zeros quickly. According to this theorem:

• If \( (x - c) \) is a factor of \( p(x) \), then \( c \) is a zero of the polynomial, meaning \( p(c) = 0 \).

In essence, the zeros represent the points where the graph of the polynomial will intersect or touch the x-axis. A polynomial of degree \( n \) can have up to \( n \) real roots but might have fewer if some roots are complex. For instance, the polynomial in the original exercise showed that \( x - \frac{1}{2} \) is a factor, telling us directly that \( \frac{1}{2} \) is a zero of the polynomial.

Polynomial Factors

Factors of a polynomial are expressions that, when multiplied together, yield the original polynomial. Understanding these factors is essential for simplifying polynomials and solving polynomial equations.
To determine if a certain expression is a factor, we apply the Factor Theorem, which connects factors directly with zeros of a polynomial. It states that:

• If \( p(x) // (x - c) \) results in \( p(c) = 0 \), then \( x - c \) is a factor of \( p(x) \).

Finding all the factors of a polynomial involves breaking it down into its simplest components, often starting by identifying one or more zeros. Factoring polynomials can help solve equations and understand their graphs. The example given in the exercise illustrates that knowing a single factor \( x - \frac{1}{2} \), we immediately know a zero, \( \frac{1}{2} \), simplifying the polynomial greatly.