Question

Use the Factor Theorem to prove that \(x+c\) is a factor of \(x^{n}+c^{n}\) if \(n \geq 1\) is an odd integer.

Short answer

Substituting x = -c into f(x) = x^n + c^n results in 0, thus x + c is a factor of f(x) when n is odd.

Step-by-step solution

Understand the Factor Theorem

The Factor Theorem states that if a polynomial function f(x) has a root at x = a, then (x - a) is a factor of f(x). To prove that x + c is a factor of f(x) = x^n + c^n, we need to show that substituting x = -c into f(x) yields f(-c) = 0.

Substitute x = -c into f(x)

Substitute x = -c into the polynomial f(x) = x^n + c^n. This gives: \[ f(-c) = (-c)^n + c^n \]

Simplify the expression

When n is an odd integer, \[ (-c)^n = -(c^n) \] Thus, \[ f(-c) = -(c^n) + c^n \]

Show that the expression equals zero

Combine the terms: \[ -c^n + c^n = 0 \] Therefore, \[ f(-c) = 0 \]

Conclude using the Factor Theorem

Since f(-c) = 0, by the Factor Theorem, x + c is a factor of the polynomial f(x) = x^n + c^n when n is an odd integer.

Key concepts

Polynomial Roots

Polynomials are expressions consisting of terms that include variables raised to various powers and coefficients. These powers are always non-negative integers.
Roots of a polynomial are values for which the polynomial evaluates to zero. For example, if f(x) = 0, then x is called a root.
Let's say we have a polynomial in the form f(x). If we find a root x = a, it implies f(a) = 0. Thus, when we substitute a into the polynomial, the result will be zero.
Finding polynomial roots is essential for various mathematical applications, including solving equations and understanding graph behavior.

Factor Theorem Proof

The Factor Theorem is a key concept in algebra that connects roots of polynomials with their factors. It states:

• If f(a) = 0 for a polynomial f(x), then (x - a) is a factor of the polynomial.

To prove that x + c is a factor of x^n + c^n when n is an odd integer, we follow these steps:

• First, we understand that x + c can be rewritten as x - (-c).
• We now substitute x = -c into the polynomial, f(x) = x^n + c^n.
• This substitution yields f(-c) = (-c)^n + c^n.
• Since n is an odd integer, (-c)^n = -c^n.
• This gives us f(-c) = -c^n + c^n, which simplifies to 0.
• Finally, by the Factor Theorem, because f(-c) = 0, x + c is a factor of x^n + c^n when n is odd.

This process shows a fundamental connection between polynomial expressions and their factors, demonstrating how a root verifies a factor.

Odd Integer Exponents

An exponent indicates how many times to multiply a base number by itself.
In general:

• Even exponents make the result positive because multiplying a negative number an even number of times cancels out the negativity.
• Odd exponents result in a negative value when the base is negative, as negative values multiplied an odd number of times stay negative.

Let's consider the example that was used in the Factor Theorem proof:
When n is odd, (-c)^n = -(c^n)
This means that raising a negative number to an odd power results in a negative outcome.
This property is crucial in proving the Factor Theorem example and understanding polynomial roots with specific conditions.

Understanding odd integer exponents helps us recognize and solve complex algebraic expressions efficiently.