Problem 302 - Precalculus

Question

If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

Step-by-Step Solution

Answer

The binomial \((x-c)\) is a factor of the polynomial \(P(x)\).

Step 1: Define the Polynomial Division

When we divide a polynomial \( P(x) \) by a binomial \( D(x) = (x - c) \), it takes the form \( P(x) = (x-c)Q(x) + R(x) \), where \( Q(x) \) is the quotient and \( R(x) \) is the remainder.

Step 2: Apply Remainder Theorem

The Remainder Theorem states that when a polynomial \( P(x) \) is divided by \( x - c \), the remainder is \( P(c) \). The problem states that the remainder is zero, so \( P(c) = 0 \).

Step 3: Conclude Divisibility

If \( P(c) = 0 \), it means that \( (x-c) \) is a factor of the polynomial \( P(x) \), because the polynomial division resulted in no remainder. Therefore, \( P(x) \) is divisible by \( (x-c) \).

Key Concepts

Remainder Theorem

The Remainder Theorem is a powerful tool that simplifies the process of evaluating polynomials at specific points. Imagine having a polynomial, say \( P(x) \), and wanting to know the remainder after dividing it by a binomial of the form \( x-c \). Rather than going through lengthy polynomial division, the Remainder Theorem provides a shortcut: calculate \( P(c) \).
This means that if you plug the value \( c \) into the polynomial and perform the computations, the result will be the remainder of the division.

• If \( P(c) = 0 \), the remainder is zero, indicating a special case we will look into further with the Factor Theorem.
• If \( P(c) eq 0 \), then \( P(c) \) is simply the remainder of the division when \( P(x) \) is divided by \( x-c \).

The beauty of this theorem lies in its simplicity. It turns a potentially complex calculation into a simple substitution.

Factor Theorem

The Factor Theorem is closely connected to the Remainder Theorem, but it focuses on identifying specific factors of polynomials. This theorem states that a binomial \( (x-c) \) is a factor of a polynomial \( P(x) \) if and only if \( P(c) = 0 \). This equivalence establishes a straightforward method for testing if a binomial is a factor.
Here's how it works:

• First, compute \( P(c) \). If you find \( P(c) = 0 \), it confirms that \( x-c \) divides the polynomial completely, with no remainder left over.
• Alternatively, if \( P(c) eq 0 \), \( x-c \) is not a factor.

Using the Factor Theorem, you can determine divisibility without conducting extensive polynomial division. This theorem not only aids in factorization but is essential in finding roots of polynomials, particularly in the context of solving equations.

Polynomial Divisibility

Polynomial divisibility is a fundamental idea in algebra that determines whether one polynomial can be evenly divided by another. If a polynomial \( P(x) \) can be divided by a binomial \( x-c \) with a remainder of zero, it confirms that \( x-c \) is a factor and \( P(x) \) is divisible by \( x-c \).
To establish divisibility:

• Start by performing polynomial division of \( P(x) \) by \( x-c \).
• Apply the Remainder Theorem to check if the remainder \( P(c) \) equals zero.
• The presence of a remainder of zero indicates complete divisibility.

Understanding when a polynomial is divisible can be crucial for finding roots and simplifying expressions. It provides insight into the structure and behavior of polynomials, highlighting relationships between their terms and factors. This concept also streamlines solving polynomial equations and aids in further algebraic manipulations.