The Remainder and Factor Theorems

Overview

The Remainder Theorem and the Factor Theorem are key concepts in polynomial division. Both of these theorems allow us to easily evaluate polynomials and find factors of polynomials.

The Remainder Theorem

The Remainder Theorem states that when a polynomial \( f(x) \) is divided by \( x - c \), the remainder is equal to \( f(c) \). In other words, you can find the remainder of the division by simply evaluating the polynomial at \( x = c \).

Example of the Remainder Theorem:

Given the polynomial \( f(x) = 2x^3 + 3x^2 - 5x + 6 \), and dividing by \( x - 1 \), we can find the remainder by evaluating \( f(1) \):

\( f(1) = 2(1)^3 + 3(1)^2 - 5(1) + 6 = 2 + 3 - 5 + 6 = 6 \).

The remainder of the division is 6.

The Factor Theorem

The Factor Theorem states that if \( f(c) = 0 \), then \( x - c \) is a factor of the polynomial \( f(x) \). This means that if the remainder of the division of a polynomial by \( x - c \) is 0, then \( x - c \) is a factor of the polynomial.

Example of the Factor Theorem:

For the polynomial \( f(x) = 2x^3 + 3x^2 - 5x + 6 \), if \( f(1) = 0 \), then \( x - 1 \) is a factor of \( f(x) \). If we check the value of \( f(1) \):

\( f(1) = 2(1)^3 + 3(1)^2 - 5(1) + 6 = 2 + 3 - 5 + 6 = 6 \).

Since \( f(1) = 6 \neq 0 \), \( x - 1 \) is not a factor of \( f(x) \).

Summary of Theorems:

  • If the remainder when dividing \( f(x) \) by \( x - c \) is \( f(c) \), then the Remainder Theorem holds.
  • If \( f(c) = 0 \), then \( x - c \) is a factor of the polynomial \( f(x) \) by the Factor Theorem.

Practice Questions

  1. Question 1: Use the Remainder Theorem to find the remainder when \( f(x) = x^3 - 4x^2 + 5x - 2 \) is divided by \( x - 2 \).
    Solution

    To find the remainder, evaluate \( f(2) \):

    \( f(2) = (2)^3 - 4(2)^2 + 5(2) - 2 = 8 - 16 + 10 - 2 = 0 \).

    The remainder is 0, so \( x - 2 \) is a factor of \( f(x) \).

  2. Question 2: Use the Factor Theorem to check if \( x + 3 \) is a factor of \( f(x) = x^3 - 2x^2 - 5x + 6 \).
    Solution

    To apply the Factor Theorem, evaluate \( f(-3) \):

    \( f(-3) = (-3)^3 - 2(-3)^2 - 5(-3) + 6 = -27 - 18 + 15 + 6 = -24 \).

    Since \( f(-3) = -24 \neq 0 \), \( x + 3 \) is not a factor of \( f(x) \).

  3. Question 3: Find the remainder when \( f(x) = 3x^4 + 5x^3 - 2x^2 + 4x - 1 \) is divided by \( x - 1 \) using the Remainder Theorem.
    Solution

    Evaluate \( f(1) \) to find the remainder:

    \( f(1) = 3(1)^4 + 5(1)^3 - 2(1)^2 + 4(1) - 1 = 3 + 5 - 2 + 4 - 1 = 9 \).

    The remainder is 9.

  4. Question 4: Use the Factor Theorem to determine if \( x - 4 \) is a factor of \( f(x) = x^3 - 6x^2 + 11x - 6 \).
    Solution

    Evaluate \( f(4) \) to check if \( x - 4 \) is a factor:

    \( f(4) = (4)^3 - 6(4)^2 + 11(4) - 6 = 64 - 96 + 44 - 6 = 6 \).

    Since \( f(4) = 6 \neq 0 \), \( x - 4 \) is not a factor of \( f(x) \).

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