Division Algorithm

Alright, let’s dive into the exciting world of Polynomial Division! ➗ It’s the final major arithmetic operation for polynomials, and while it might seem a bit daunting at first, it’s very much like the long division you already know, just with some algebraic twists! 😉

What is Polynomial Division? 🤔

Just as you can divide whole numbers, for example, $10 \div 3$ results in $3$ with a remainder of $1$, you can divide one polynomial by another. This process aims to break down a more complex polynomial (the dividend) into simpler parts using another polynomial (the divisor).

The fundamental relationship that defines division remains the same for polynomials: Dividend = Quotient × Divisor + Remainder

It’s important to note that, unlike addition, subtraction, or multiplication of polynomials (which always result in another polynomial), the outcome of dividing two polynomials will not always be another polynomial if there’s a non-zero remainder. In such cases, the result is expressed as a rational function.

Key Terms to Know 📚

To understand polynomial division, let’s get familiar with the players involved:

• Dividend (P(x)): This is the polynomial being divided, much like the number you put inside the division symbol. It’s the “numerator” in a fractional expression.
• Divisor (Q(x)): This is the polynomial you are dividing by, the one on the outside of the division symbol. It’s the “denominator”.
• Quotient (H(x)): This is the main result of the division, indicating how many times the divisor “fits into” the dividend. It’s the polynomial part of your answer.
• Remainder (R(x)): This is the polynomial left over after the division, once no further full divisions are possible. Think of it like the “1” when you divide 10 by 3.

Crucial Conditions! 🚨

Before you start dividing, there are a couple of very important rules to keep in mind:

1. Degree Relationship 📏: For polynomial division (to find a polynomial quotient) to be possible, the degree of the dividend must be greater than or equal to the degree of the divisor.
   • What if the Dividend’s Degree is Smaller? If the degree of the dividend $P(x)$ is less than the degree of the divisor $Q(x)$, then the division is not typically “possible” in the sense of yielding a polynomial quotient. In this specific scenario, the quotient is the zero polynomial (0), and the remainder is simply the dividend itself (P(x)). For instance, you wouldn’t typically perform long division for $4 \div (2x+1)$ to get a polynomial result, because $4$ has degree $0$ and $2x+1$ has degree $1$.
2. Remainder’s Degree 👇: The degree of the remainder R(x) must always be strictly less than the degree of the divisor Q(x). If the remainder’s degree is still equal to or greater than the divisor’s, it means you haven’t finished the division yet!
3. Zero Polynomial’s Degree 🚫: It’s a special case, but the degree of the zero polynomial (0) is always undefined. This comes into play if your division results in a remainder of 0.

The Polynomial Long Division Algorithm (Step-by-Step) 🪜

This is the most common and systematic method for dividing polynomials. Let’s break it down:

1. Arrange and Fill Gaps 📝:
   • Write both the dividend and the divisor in descending order of their degrees (highest power first, down to the constant term).
   • If any powers of the variable are missing in the dividend, add them with a coefficient of zero as placeholders (e.g., $x^4 + 2x^2 + 3x + 2$ should be written as $x^4 + 0x^3 + 2x^2 + 3x + 2$). This helps keep terms aligned.
2. Divide Leading Terms ➗:
   • Divide the first term (leading monomial) of the current dividend by the first term (leading monomial) of the divisor.
   • The result of this division is the first term of your quotient.
3. Multiply and Subtract ✖️➖:
   • Take the term you just found in the quotient (from Step 2) and multiply it by the entire divisor.
   • Write this product directly below the current dividend, aligning like terms.
   • Subtract this entire product from the current dividend. Be very careful with signs! It’s often helpful to change the signs of all terms in the product and then add.
4. Bring Down and Repeat or Terminate 🔁🛑:
   • Bring down the next term (or terms) from the original dividend to form your new current dividend.
   • Now, look at this new polynomial (your new “dividend”):
     • If its degree is still greater than or equal to the degree of the divisor, then repeat from Step 2 with this new polynomial.
     • If its degree is strictly less than the degree of the divisor, then this polynomial is your remainder (R(x)), and the process stops.

Let’s Dive into an Example! 🚀

Let’s divide $P(x) = x^4 + 2x^2 + 3x + 2$ by $Q(x) = x^2 + x + 1$.

Step 1: Arrange and Fill Gaps 📝 The dividend needs a placeholder for $x^3$: $P(x) = x^4 + 0x^3 + 2x^2 + 3x + 2$ $Q(x) = x^2 + x + 1$

                  <-- Quotient (will go here)
        _________________
x² + x + 1 | x⁴ + 0x³ + 2x² + 3x + 2   <-- Dividend

First Iteration:

• Step 2: Divide Leading Terms ➗ ($x^4 \div x^2 = x^2$). This is the first term of our quotient.
• Step 3: Multiply and Subtract ✖️➖ ($x^2 \times (x^2 + x + 1) = x^4 + x^3 + x^2$)

        x²          <-- First term of Quotient
      _________________
x² + x + 1 | x⁴ + 0x³ + 2x² + 3x + 2
       -(x⁴ + x³ + x²)  <-- Product of x² and Divisor, then subtracted
       _________________
             -x³ + x² + 3x + 2   <-- New Dividend

• Step 4: Bring Down and Repeat 🔁 The degree of $-x^3 + x^2 + 3x + 2$ (which is 3) is still greater than or equal to the degree of $x^2 + x + 1$ (which is 2). So, we repeat.

Second Iteration:

• Step 2: Divide Leading Terms ➗ (The leading term of our new dividend, $-x^3$, divided by $x^2$ is $-x$). This is the next term of our quotient.
• Step 3: Multiply and Subtract ✖️➖ ($-x \times (x^2 + x + 1) = -x^3 - x^2 - x$)

        x² - x      <-- Quotient so far
      _________________
x² + x + 1 | x⁴ + 0x³ + 2x² + 3x + 2
       -(x⁴ + x³ + x²)
       _________________
             -x³ + x² + 3x + 2   
           -(-x³ - x² - x)   <-- Product of -x and Divisor, then subtracted
           _________________
                   2x² + 4x + 2   <-- New Dividend

• Step 4: Bring Down and Repeat 🔁 The degree of $2x^2 + 4x + 2$ (which is 2) is still equal to the degree of $x^2 + x + 1$ (which is 2). So, we repeat.

Third Iteration:

• Step 2: Divide Leading Terms ➗ (The leading term of our new dividend, $2x^2$, divided by $x^2$ is $2$). This is the final term of our quotient.
• Step 3: Multiply and Subtract ✖️➖ ($2 \times (x^2 + x + 1) = 2x^2 + 2x + 2$)

        x² - x + 2  <-- Final Quotient
      _________________
x² + x + 1 | x⁴ + 0x³ + 2x² + 3x + 2
       -(x⁴ + x³ + x²)
       _________________
             -x³ + x² + 3x + 2
           -(-x³ - x² - x)
           _________________
                   2x² + 4x + 2
                 -(2x² + 2x + 2)  <-- Product of 2 and Divisor, then subtracted
                 _________________
                         2x       <-- Remainder

• Step 4: Terminate! 🛑 The degree of $2x$ (which is 1) is strictly less than the degree of $x^2 + x + 1$ (which is 2). The division stops.

Final Answer: For $P(x) = x^4 + 2x^2 + 3x + 2$ and $Q(x) = x^2 + x + 1$: The Quotient is $H(x) = x^2 - x + 2$. The Remainder is $R(x) = 2x$.

This means we can write: $\frac{x^4 + 2x^2 + 3x + 2}{x^2 + x + 1} = x^2 - x + 2 + \frac{2x}{x^2 + x + 1}$. As observed, since the remainder is not zero, the result is not solely a polynomial but a rational function.

Practice Questions for You! 🧠

Question 1: Divide the polynomial $P(x) = 15x^5 - 10x^3 + 35x^2$ by the monomial $Q(x) = 5x^2$. Is the result a polynomial?

Question 2: Divide $A(x) = x^3 + 5x^2 + 6x + 2$ by $B(x) = x + 2$.

Question 3: What is the maximum possible degree of the remainder when a polynomial is divided by a divisor of degree 5?

Solutions! ✨

Solution 1: Divide $P(x) = 15x^5 - 10x^3 + 35x^2$ by $Q(x) = 5x^2$.

When dividing by a monomial (a polynomial with a single term), you can simply divide each term of the dividend by the monomial separately:

$\frac{15x^5 - 10x^3 + 35x^2}{5x^2}$ $= \frac{15x^5}{5x^2} - \frac{10x^3}{5x^2} + \frac{35x^2}{5x^2}$ $= (15 \div 5)x^{5-2} - (10 \div 5)x^{3-2} + (35 \div 5)x^{2-2}$ $= 3x^3 - 2x^1 + 7x^0$ $= \mathbf{3x^3 - 2x + 7}$

Result: The quotient is $3x^3 - 2x + 7$, and the remainder is $\mathbf{0}$. Is the result a polynomial? Yes, because the remainder is $0$.

Solution 2: Divide $A(x) = x^3 + 5x^2 + 6x + 2$ by $B(x) = x + 2$.

Step 1: Arrange and Fill Gaps 📝 Both polynomials are already in descending order, and there are no missing terms.

                  <-- Quotient (will go here)
        _________________
x + 2   | x³ + 5x² + 6x + 2

First Iteration:

• Step 2: Divide Leading Terms ➗ ($x^3 \div x = x^2$).
• Step 3: Multiply and Subtract ✖️➖ ($x^2 \times (x + 2) = x^3 + 2x^2$)

        x²
      _________________
x + 2   | x³ + 5x² + 6x + 2
       -(x³ + 2x²)
       _________________
             3x² + 6x + 2   <-- New Dividend

• Step 4: Bring Down and Repeat 🔁 The degree of $3x^2 + 6x + 2$ (which is 2) is greater than or equal to the degree of $x + 2$ (which is 1). So, we repeat.

Second Iteration:

• Step 2: Divide Leading Terms ➗ ($3x^2 \div x = 3x$).
• Step 3: Multiply and Subtract ✖️➖ ($3x \times (x + 2) = 3x^2 + 6x$)

        x² + 3x
      _________________
x + 2   | x³ + 5x² + 6x + 2
       -(x³ + 2x²)
       _________________
             3x² + 6x + 2
           -(3x² + 6x)
           _____________
                     0x + 2   <-- New Dividend (or just 2)

• Step 4: Bring Down and Repeat 🔁 The degree of $2$ (which is 0) is less than the degree of $x + 2$ (which is 1). This means we have found our remainder.

Final Answer: For $A(x) = x^3 + 5x^2 + 6x + 2$ and $B(x) = x + 2$: The Quotient is $H(x) = \mathbf{x^2 + 3x}$. The Remainder is $R(x) = \mathbf{2}$.

This means we can write: $\frac{x^3 + 5x^2 + 6x + 2}{x + 2} = x^2 + 3x + \frac{2}{x + 2}$.

Solution 3: What is the maximum possible degree of the remainder when a polynomial is divided by a divisor of degree 5?

As per the crucial conditions of polynomial division, the degree of the remainder R(x) must always be strictly less than the degree of the divisor Q(x).

If the divisor has a degree of 5, then the remainder’s degree must be less than 5. Therefore, the maximum possible degree of the remainder is 4. ✅