Question

Use the Factor Theorem to prove that \(x-c\) is a factor of \(x^{n}-c^{n}\) for any positive integer \(n\).

Short answer

Using the Factor Theorem, \(x - c\) is a factor of \(x^n - c^n\) since \(f(c) = 0\).

Step-by-step solution

State the Factor Theorem

The Factor Theorem states that if a polynomial function \(f(x)\) has a root at \(x = c\), then \(x - c\) is a factor of \(f(x)\). In other words, \(f(c) = 0\) if and only if \(x - c\) is a factor of \(f(x)\).

Define the Polynomial

Consider the polynomial \(f(x) = x^{n} - c^{n}\). According to the problem, you need to show that \(x - c\) is a factor of \(x^n - c^n\).

Evaluate the Polynomial at \(x=c\)

Substitute \(x = c\) into the polynomial: \(f(c) = c^{n} - c^{n}\).

Simplify the Expression

Simplify the expression \(f(c) = c^{n} - c^{n} = 0\). This shows that \(f(c) = 0\).

Apply the Factor Theorem

Since \(f(c) = 0\), by the Factor Theorem, \(x - c\) is a factor of \(f(x) = x^{n} - c^{n}\).

Key concepts

polynomial function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. A single-variable polynomial function can be expressed in the form of \( f(x) = a_n x^n + a_{n-1} x^{n-1} + \.\.\. + a_1 x + a_0 \), where \( a_n, a_{n-1}, ..., a_0 \) are constants.

Polynomial functions are widely used due to their simplicity and smooth graphical properties.

• The degree of a polynomial is the highest power of the variable in the function. For example, in \( f(x) = 2x^3 + 3x^2 + x + 5 \), the degree is 3.
• The coefficients are the constants multiplying each term. In the previous example, the coefficients are 2, 3, 1, and 5.
• The leading coefficient is the coefficient of the term with the highest power. For the example above, it's 2.

Polynomial functions are important in calculus since they are easy to differentiate and integrate. Their smooth curves also make them significant in modeling real-world situations.

roots of polynomials

The roots (or zeros) of a polynomial are the values of \( x \) that make the polynomial equal to zero. For a polynomial function \( f(x) \), if \( f(c) = 0 \), then \( c \) is a root of the polynomial.

Roots can be real or complex and play a crucial role in understanding polynomial behavior.

• **Real roots** are the values where the polynomial intersects the x-axis on a graph.
• **Complex roots** occur in conjugate pairs and do not appear on the real-number graph.

Finding the roots is essential in various applications such as solving equations and analyzing functions. One useful approach is the Factor Theorem, which states that if \( c \) is a root of the polynomial \( f(x) \), then \( x - c \) is a factor of \( f(x) \). This simplifies the process of factorizing polynomials and understanding their behavior.

factorization

Factorization is the process of breaking down a polynomial into a product of simpler polynomials that, when multiplied, give the original polynomial. It often involves finding the roots of the polynomial first.

For example, to factorize \( x^2 - 5x + 6 \), you would find its roots.

• Identifying roots 2 and 3: \( f(x) = (x-2)(x-3) \)

Using the Factor Theorem can significantly simplify this process. In our example, by showing that \( x - c \) is a factor of \( x^n - c^n \) for any positive integer \( n \), you leverage the theorem to directly infer factors based on known roots.

Factorization is useful in solving polynomial equations, simplifying expressions, and in calculus for finding limits and deriving formulas.