Introduction
Hello, Sobat Raita! Welcome to our comprehensive guide on how to prove the division algorithm for polynomials using mathematical induction. This method is a powerful technique used in polynomial algebra to demonstrate that any polynomial can be divided by another polynomial with a remainder that is either zero or of lower degree than the divisor.
In this article, we will provide a step-by-step explanation of the proof along with detailed examples to help you understand the concept clearly. So, grab a cup of coffee and let’s dive into the fascinating world of polynomials!
Understanding the Division Algorithm
Polynomial Division
Polynomial division is similar to long division for numbers. Given two polynomials, $f(x)$ and $g(x)$, where $g(x) ≠ 0$, the division algorithm states that we can express $f(x)$ as a quotient $q(x)$ and a remainder $r(x)$ such that:
$$f(x) = g(x)q(x) + r(x)$$
where the degree of $r(x)$ is less than the degree of $g(x).
Mathematical Induction
Mathematical induction is a mathematical technique used to prove statements that hold for all natural numbers. It involves two steps:
- Base Case: Prove the statement for the smallest natural number (usually 1).
- Inductive Step: Assume the statement is true for some $k$ and prove that it is also true for $k+1$.
Proof Using Induction
Base Case (n = 1)
When the degree of $f(x)$ is less than the degree of $g(x)$, we can perform polynomial division directly, and the remainder will be either zero or a constant. This satisfies the division algorithm, so the base case holds.
Inductive Step (Assume n = k)
Assume that for some $k$, any polynomial $f(x)$ of degree $k$ can be expressed as $f(x) = g(x)q(x) + r(x)$, where $r(x)$ is either zero or of degree less than $g(x)$.
Prove for n = k + 1
We need to show that any polynomial $f(x)$ of degree $k+1$ can also be expressed in the form $f(x) = g(x)q(x) + r(x)$. Consider a polynomial $f(x)$ of degree $k+1$. We can write $f(x)$ as:
$$f(x) = x^{k+1} + a_kx^k + … + a_1x + a_0$$
where $a_i$ are coefficients.
We can rewrite $f(x)$ as:
$$f(x) = x(x^k + a_{k-1}x^{k-1} + … + a_1x + a_0) + a_0$$
By the induction hypothesis, we know that $x^k + a_{k-1}x^{k-1} + … + a_1x + a_0$ can be expressed as a quotient $q(x)$ and a remainder $r(x)$ when divided by $g(x)$. Therefore, we have:
$$f(x) = x(g(x)q(x) + r(x)) + a_0$$
$$f(x) = g(x)(xq(x)) + (xr(x) + a_0)$$
Let $q'(x) = xq(x)$ and $r'(x) = xr(x) + a_0$. Then, we have:
$$f(x) = g(x)q'(x) + r'(x)$$
where $r'(x)$ is either zero or of degree less than $g(x)$. This completes the inductive step.
Examples of Division Algorithm
Example 1: Polynomials with Integer Coefficients
Prove that $x^3 – 2x^2 + 3x – 4$ is divisible by $x – 2$.
Using polynomial division, we get:
$$\begin{array}{c}
\qquad\qquad\qquad\quad\ x^2\\\
x-2\qquad\overline{)x^3-2x^2+3x-4} \\\
\qquad\qquad\qquad\ x^3-2x^2 \\\
\qquad\qquad\qquad\qquad\qquad 3x-4 \\\
\qquad\qquad\qquad\qquad\qquad 3x-6 \\\
\qquad\qquad\qquad\qquad\qquad\qquad 2
\end{array}$$
Therefore, $x^3 – 2x^2 + 3x – 4 = (x – 2)(x^2) + 2$, which proves the divisibility.
Example 2: Polynomials with Real Coefficients
Prove that $x^4 – 1$ is divisible by $x^2 + 1$.
Using polynomial division, we get:
$$\begin{array}{c}
\qquad\qquad\qquad\quad\ x^2-1\\\
x^2+1\qquad\overline{)x^4-1} \\\
\qquad\qquad\qquad\ x^4+1 \\\
\qquad\qquad\qquad\qquad\qquad -2 \\\
\qquad\qquad\qquad\qquad\qquad -2-1 \\\
\qquad\qquad\qquad\qquad\qquad\qquad 1
\end{array}$$
Therefore, $x^4 – 1 = (x^2 + 1)(x^2 – 1) + 1$, which proves the divisibility.
Table: Division Algorithm for Polynomials
Table 1: Division Algorithm for Polynomials
Frequently Asked Questions (FAQs)
What is the division algorithm?
The division algorithm for polynomials states that any polynomial can be divided by another non-zero polynomial, resulting in a quotient and a remainder of lower degree.
How is mathematical induction used to prove the division algorithm?
Mathematical induction involves proving a statement for a base case and then assuming it is true for an arbitrary natural number $k$ to prove it for $k+1$.
Can the division algorithm be used for polynomials with complex coefficients?
Yes, the division algorithm holds for polynomials with any type of coefficients, including complex coefficients.
What are the limitations of the polynomial division algorithm?
The polynomial division algorithm requires the divisor to be non-zero. It cannot be used to divide by the zero polynomial.
How is the division algorithm used in practice?
The division algorithm is used in various applications, such as finding remainders, solving polynomial equations, and checking for divisibility.
What is the Euclidean algorithm?
The Euclidean algorithm is an extension of the division algorithm that provides a step-by-step procedure to find the greatest common divisor of two polynomials.
How does the division algorithm relate to synthetic division?
Synthetic division is a simplified method of polynomial division that is based on the division algorithm.
What are some real-world applications of the division algorithm?
The division algorithm is used in cryptography, error-correcting codes, and computer science.
How can I learn more about the division algorithm for polynomials?
There
