Problem 62 - Precalculus with Calculus Previews

Question

Suppose \(f\) and \(g\) are polynomial functions. If \(c\) is a zero of \(f\) and of \(g\), then show that \(c\) a zero of \(f+g\) and \(f g\).

Step-by-Step Solution

Answer

Yes, if \(c\) is a zero of both polynomials, it is a zero of \(f+g\) and \(fg\).

Step 1: Understanding Zeroes of Polynomials

A zero of a polynomial function means that substituting that value into the function gives us zero. If \( c \) is a zero of function \( f \), then \( f(c) = 0 \). Similarly, if \( c \) is a zero of function \( g \), then \( g(c) = 0 \). We must show that \( c \) is also a zero for \( f+g \) and \( f g \).

Step 2: Zero of the Sum of Polynomials

To show that \( c \) is a zero of \( f+g \), consider the function \( (f+g)(x) = f(x) + g(x) \). Therefore, \((f+g)(c) = f(c) + g(c) \). Since \( f(c) = 0 \) and \( g(c) = 0 \), it follows that \( (f+g)(c) = 0 + 0 = 0 \). Thus, \( c \) is a zero of \( f+g \).

Step 3: Zero of the Product of Polynomials

To show that \( c \) is a zero of \( f g \), consider the function \( (f g)(x) = f(x)g(x) \). Therefore, \((f g)(c) = f(c)g(c) \). Since \( f(c) = 0 \) and \( g(c) = 0 \), it follows that \( (f g)(c) = 0 \times 0 = 0 \). This shows that \( c \) is also a zero of \( f g \).

Key Concepts

Zeros of Polynomials

Zeros of polynomial functions are the values for which the polynomial evaluates to zero. If you substitute a zero for a polynomial, you will always end up with a result of zero. For a polynomial \( f(x) \), if \( c \) is a zero, this means \( f(c) = 0 \).
Similarly, for another polynomial \( g(x) \), if \( c \) is also a zero, then \( g(c) = 0 \). This foundational concept helps us explore interactions between polynomials, such as their sums and products.
An easy way to spot zeros in a polynomial is to find the roots or solutions to the equation \( f(x) = 0 \). These roots can often be found using methods like factoring, using the quadratic formula in case of quadratic polynomials, or employing numerical methods for polynomials of higher degrees.

Sums of Polynomials

When you add two polynomials, each term from both polynomials is added together. If \( f(x) \) and \( g(x) \) are two polynomials, their sum is written as \((f+g)(x) = f(x) + g(x)\).
For example, if \( f(x) = 2x^2 + 3x + 1 \) and \( g(x) = -x^2 + 4x + 3 \), you simply add the like terms: \((f+g)(x) = (2x^2 - x^2) + (3x + 4x) + (1 + 3) = x^2 + 7x + 4\).
Finding zeros in sums is straightforward if you already know the zeros of the individual polynomials. For instance, if \( c \) is a zero for both \( f \) and \( g \), then substituting \( c \) into the sum \((f+g)(c)\) gives you:

• \( f(c) = 0 \) implies that substituting \( c \) in \( f(x) \) results in zero.
• \( g(c) = 0 \) implies that substituting \( c \) in \( g(x) \) results in zero too.

This naturally means \((f+g)(c) = 0 + 0 = 0\), making \( c \) a zero of the sum.

Products of Polynomials

The product of two polynomials is quite different from their sum. When you multiply \( f(x) \) and \( g(x) \), their product \((f g)(x)\) means multiplying every term in \( f(x) \) with every term in \( g(x) \).
For instance, with \( f(x) = x+1 \) and \( g(x) = x-1 \), the product will be \( (x+1)(x-1) = x^2 - 1 \).
The beauty of polynomial products is that if \( c \) is a zero of each of \( f(x) \) and \( g(x) \), then it will also be a zero of their product \((f g)(x)\).
To understand this, remember that \( f(c) \) and \( g(c) \) are both zero. So in the product, \((f g)(c) = f(c)g(c) = 0 \times 0 = 0\), clearly showing \( c \) is a zero of \( (f g)(x) \).
This concept is crucial in polynomial algebra, especially in finding common solutions across multiple polynomial equations.

Polynomial Algebra

Polynomial algebra is a branch of mathematics dealing with the operations on polynomial expressions. Understanding the behavior of sums and products of polynomials is fundamental in algebra.
Key operations in polynomial algebra include:

• Addition: Combining like terms of two or more polynomials.
• Subtraction: Subtracting matching terms between polynomials.
• Multiplication: Distributing each term from one polynomial to every term in another, using the distributive property.
• Division: Breaking down polynomials into simpler fractions, often using polynomial long division.

Recognizing the significance of zeros is also vital. They are often used to factor polynomials, simplify expressions, and solve polynomial equations. Understanding these zeros helps in constructing graphs, analyzing polynomial functions' behavior, and solving problems.
The overarching goal in polynomial algebra is to manipulate polynomials to solve equations, prove equalities, or simplify expressions in ways that reveal the deeper structure or meaning of the algebraic form.