Remainder, Factor Theorems, Ruffini’s Rule, and Polynomial Algebra

Remainder, Factor Theorems, Ruffini’s Rule, and Polynomial Algebra §

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Remainder Theorem: §

1. Definition: When a polynomial $P(x)$ is divided by a linear divisor $x-a$, the remainder is equal to $P(a)$.

Factor Theorem: §

1. Definition: For a polynomial $P(x)$, if $P(a)=0$, then $x-a$ is a factor of $P(x)$.

2. Connection with Remainder Theorem: The Factor Theorem is essentially a special case of the Remainder Theorem. If the remainder when $P(x)$ is divided by $x-a$ is zero, then $x-a$ is a factor of $P(x)$.

Ruffini’s Rule: §

Ruffini’s Rule, also known as Ruffini’s Horner’s method, is a quick polynomial division technique when dividing a polynomial by a linear divisor of the form $x-c$. It’s especially useful for synthetic division.

Let’s go step-by-step:

Ruffini’s Rule Procedure

1. Setup:

   • Write down the polynomial you want to divide. Let’s call this polynomial $P(x)$.
   • Identify the divisor. It should be in the form $x-c$ or $x+c$.
   • List the coefficients of $P(x)$ in descending order of the degree of each term.

2. Initialize the table:

   • Draw a horizontal line and a vertical line to create a small ‘L’ shape.
   • Above the horizontal line and to the left of the vertical line, write the value of $c$.
   • Next to $c$ and above the horizontal line, write down the coefficients of $P(x)$ in order, starting from the highest degree. If a degree is missing, place a 0 for that coefficient. This creates the first row of numbers.

3. First Entry:

   • Bring down the leading coefficient (the leftmost number of the first row) as it is. Write it below the horizontal line to start a new row.

4. Multiplication and Addition:

   • Multiply the number you just wrote below the line by $c$ (the value you placed at the corner of the ‘L’ shape). Write the result below the next coefficient in the first row.
   • Add the coefficient from the first row and the result you just obtained. Write this sum below the line, creating the next entry in the second row.
   • Continue this process of multiplying, then adding, for all the coefficients in the first row.

5. Result:

   • Once you’ve gone through all the coefficients, the numbers in the second row represent the coefficients of the quotient polynomial in descending order.
   • The rightmost number in the second row represents the remainder.

6. Interpret the Answer:

   • The coefficients obtained in the second row (except for the last number) represent the quotient polynomial.
   • If the remainder is 0, it means $P(x)$ is completely divisible by $x-c$. If the remainder is not 0, then it’s the remainder when $P(x)$ is divided by $x-c$.

Example: Let’s illustrate with an example.

Let’s say we are dividing $P(x)=x^3-6x^2+11x-6$ by $x-1$:

1. We set up our polynomial and identify our divisor as $x-1$. So, $c=1$.
2. We write 1 at the corner of our ‘L’ shape and list our coefficients as 1, -6, 11, and -6.
3. The leading coefficient, 1, is brought straight down.
4. We multiply this 1 by $c$, which gives 1. We then add this to the next coefficient, -6, giving -5.
5. We continue this process. Multiplying -5 by 1 gives -5, which when added to 11 gives 6. Then, multiplying 6 by 1 and adding to -6 gives 0.
6. Our final row is 1, -5, 6, and 0. This means our quotient polynomial is $x^2-5x+6$ and our remainder is 0.

The procedure described above may sound a bit lengthy due to the verbose nature of the explanation, but in practice, Ruffini’s Rule is a quick and efficient method to perform polynomial division by a linear divisor.

Tabular Example The table is set up as shown below

(2)	3	0	-6	2
	0
	3
Then slowly we apply the procedure as described above.

(2)	3	0	-6	2
	0	6	12	12
	3	6	6	14
Thus $q(x)=3x^2+6x+6$, $r(x)=\frac{14}{x-2}$.

Application to Factorising $x^n\pm a^n$: §

1. Factorisation of $x^n-a^n$: $x^n-a^n$ can be factored as $(x-a)(x^{n-1}+x^{n-2}a+x^{n-3}{a}^2+\cdots +x{a}^{n-2}+a^{n-1})$

2. Factorisation of $x^n+a^n$ for odd $n$: $x^n+a^n$ can be factored using similar principles as $x^n-a^n$, but the signs within the factors will alternate.

$$p(x)=x^3+2x^2-x-3=a_1(x-2{)}^3+a_2(x-2{)}^2+a_3(x-2)+a_4$$

$$p(x)=(x-2)q(x)+a_4$$

$$\text{for }q(x)=a_1(x-2{)}^2+a_2(x-2)+a_3$$

We can continue this, slowly decomposing the polynomial $p(x)$. This proves how we can find the coefficients (if continued).

Expanding a Polynomial in Terms of $x-a$: §

Given a polynomial $P(x)$, its expansion in terms of $x-a$ will result in a polynomial in which every term will be of the form $c_k(x-a{)}^k$, where $c_k$ are coefficients determined by the polynomial and the value of $a$.

GCD for Polynomials: §

1. Definition: The greatest common divisor (GCD) of two polynomials is the highest-degree polynomial that divides both of the given polynomials without leaving a remainder.

2. Procedure: The GCD of polynomials can be found using the polynomial division method or the extended Euclidean algorithm.

The Extended Euclidean Algorithm for Polynomials: §

1. Definition: This algorithm is a polynomial version of the extended Euclidean algorithm for integers. It helps find the GCD of two polynomials and represents the GCD as a linear combination of the two polynomials.

2. Procedure: The method involves repeated polynomial division, just as the standard extended Euclidean algorithm does with integers.

Bézout’s Lemma for Polynomials: §

1. Definition: For polynomials $P(x)$ and $Q(x)$ with a GCD $D(x)$, there exist polynomials $U(x)$ and $V(x)$ such that: $P(x)U(x)+Q(x)V(x)=D(x)$

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Historical Context: §

1. Remainder and Factor Theorems: These theorems, foundational to polynomial algebra, have been known since ancient times and have played a pivotal role in algebraic computations and problem-solving.

2. Ruffini’s Rule: Named after Paolo Ruffini, an Italian mathematician who made significant contributions to algebra.

3. Bézout’s Lemma: Named after Étienne Bézout, a French mathematician who first stated and proved this lemma for integers. Its adaptation to polynomials has been instrumental in algebraic computations.

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Sample Exam Questions: §

1. Remainder Theorem: Given the polynomial $P(x)=2x^3-3x^2+4x-5$, determine the remainder when $P(x)$ is divided by $x-2$.

2. Factor Theorem: Using the Factor Theorem, determine whether $x-3$ is a factor of $Q(x)=x^3-6x^2+11x-6$.

3. Ruffini’s Rule: Use Ruffini’s Rule to divide $P(x)=x^3-2x^2-4x+8$ by $x+1$.

4. Polynomial Expansion: Expand the polynomial $f(x)=(x-2)(x-3)(x-4)$ in terms of $x-2$.

5. Extended Euclidean Algorithm: Use the extended Euclidean algorithm to find the GCD of the polynomials $A(x)=x^4-1$ and $B(x)=x^3+x^2+x+1$.

6. Conceptual Question: Describe the importance of Bézout’s Lemma in polynomial algebra and its connection to the extended Euclidean algorithm.