Polynomials Division

Right then! We’ve mastered the art of adding and subtracting polynomials, and last time, we unravelled the secrets of multiplication. Now, let’s tackle the final frontier of polynomial algebra: Division! ➗ It’s a bit like long division with numbers, but with variables thrown into the mix. Don’t worry, we’ll make it as easy as pie (polynomial pie, of course! 🥧).

What is Polynomial Division? 🤔

Just like when you divide numbers (e.g., $10 \div 3$ gives $3$ with a remainder of $1$), polynomial division involves splitting one polynomial (the dividend) by another (the divisor) to find a quotient and a remainder.

The fundamental relationship in division is: Dividend = Quotient × Divisor + Remainder

The main goal is to break down a complex polynomial into simpler parts. Crucially, unlike addition, subtraction, or multiplication, the result of dividing two polynomials will not always be another polynomial.

Key Terminology to Remember 📚

• Dividend (P(x)): The polynomial being divided (the numerator).
• Divisor (Q(x)): The polynomial that divides the dividend (the denominator).
• Quotient (H(x)): The result of the division, much like the ‘3’ in $10 \div 3$.
• Remainder (R(x)): The polynomial left over after the division, like the ‘1’ in $10 \div 3$. The degree of the remainder must always be strictly less than the degree of the divisor.

The Rules of the Game (Important Conditions!) 💡

When dividing polynomials, there’s a vital condition regarding their degrees: The degree of the dividend must be greater than or equal to the degree of the divisor.

• If the degree of the dividend is less than the degree of the divisor, then the division (in terms of finding a polynomial quotient) isn’t directly possible. In this scenario, the quotient is the zero polynomial (0), and the remainder is simply the dividend itself. For example, you can’t really divide $4$ by $2x+1$ to get a polynomial result because $4$ has degree $0$ and $2x+1$ has degree $1$.

Step-by-Step: The Division Algorithm (Long Division) 👷‍♀️

This is the most common method for polynomial division. It systematically finds the quotient and remainder.

Here are the steps involved, often called the Division Algorithm:

1. Arrange and Fill Gaps 📏: Arrange the terms of both the dividend and the divisor in descending order of their degrees. If any powers of the variable are missing in the dividend, add them with a coefficient of zero as a placeholder (e.g., $x^4 + 2x^2 + 3x + 2$ should be thought of as $x^4 + 0x^3 + 2x^2 + 3x + 2$).
2. Divide Leading Terms ➗: Divide the first term (leading monomial) of the current dividend by the first term (leading monomial) of the divisor. This result is the next term of your quotient.
3. Multiply and Subtract ✖️➖: Multiply the term you just found in the quotient (from Step 2) by the entire divisor. Write this product below the current dividend, aligning like terms. Then, subtract this product from the current dividend.
4. Repeat or Terminate 🔁🛑: Look at the resulting polynomial after subtraction (this is 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 as your dividend.
   • If its degree is less than the degree of the divisor, then this polynomial is your remainder (R(x)), and the process stops.

Let’s Work Through 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 $P(x) = x^4 + 0x^3 + 2x^2 + 3x + 2$ $Q(x) = x^2 + x + 1$

        x² - x + 2          <-- Quotient (H(x))
      _________________
x²+x+1 | x⁴ + 0x³ + 2x² + 3x + 2   <-- Dividend (P(x))

Step 2: Divide Leading Terms ($x^4 / x^2 = x^2$) Step 3: Multiply and Subtract ($x^2 \times (x^2 + x + 1) = x^4 + x^3 + x^2$)

        x²
      _________________
x²+x+1 | x⁴ + 0x³ + 2x² + 3x + 2
       -(x⁴ + x³ + x²)
       _________________
             -x³ + x² + 3x + 2   <-- New Dividend

Step 4: Repeat (Degree of -x³ is 3, which is >= Degree of x² (2))

Step 2: Divide Leading Terms ($-x^3 / x^2 = -x$) Step 3: Multiply and Subtract ($-x \times (x^2 + x + 1) = -x^3 - x^2 - x$)

        x² - x
      _________________
x²+x+1 | x⁴ + 0x³ + 2x² + 3x + 2
       -(x⁴ + x³ + x²)
       _________________
             -x³ + x² + 3x + 2
           -(-x³ - x² - x)
           _________________
                   2x² + 4x + 2   <-- New Dividend

Step 4: Repeat (Degree of 2x² is 2, which is >= Degree of x² (2))

Step 2: Divide Leading Terms ($2x^2 / x^2 = 2$) Step 3: Multiply and Subtract ($2 \times (x^2 + x + 1) = 2x^2 + 2x + 2$)

        x² - x + 2
      _________________
x²+x+1 | x⁴ + 0x³ + 2x² + 3x + 2
       -(x⁴ + x³ + x²)
       _________________
             -x³ + x² + 3x + 2
           -(-x³ - x² - x)
           _________________
                   2x² + 4x + 2
                 -(2x² + 2x + 2)
                 _________________
                         2x       <-- Remainder (R(x))

Step 4: Terminate (Degree of 2x is 1, which is < Degree of x² (2))

So, 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 that $\frac{x^4 + 2x^2 + 3x + 2}{x^2 + x + 1} = x^2 - x + 2 + \frac{2x}{x^2 + x + 1}$. As you can see, the result is not purely a polynomial, because of the remainder term. This highlights that polynomial division doesn’t always yield another polynomial.

If the remainder is $0$, it means the divisor is a factor of the dividend.

Practice Questions! 📝

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 to Practice Questions ✅

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

When dividing by a monomial, you can divide each term of the dividend by the monomial individually: $\frac{15x^5}{5x^2} - \frac{10x^3}{5x^2} + \frac{35x^2}{5x^2}$ $= (15/5)x^{5-2} - (10/5)x^{3-2} + (35/5)x^{2-2}$ $= 3x^3 - 2x^1 + 7x^0$ $= 3x^3 - 2x + 7$

Result: The quotient is $3x^3 - 2x + 7$, and the remainder is $0$. Is it a polynomial? Yes, because the remainder is $0$. This is also an example of how multiplication is the reverse of division, as we saw $2x^3 \times (x^2 + x + 1) = 2x^5 + 2x^4 + 2x^3$ (from our previous conversation).

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

Step 1: Arrange and Fill Gaps (No missing terms here!)

        x² + 3x + 0          <-- Quotient (H(x))
      _________________
x+2   | x³ + 5x² + 6x + 2   <-- Dividend (A(x))

Step 2: Divide Leading Terms ($x^3 / x = x^2$) 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: Repeat (Degree of 3x² is 2, which is >= Degree of x (1))

Step 2: Divide Leading Terms ($3x^2 / x = 3x$) 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 (just 2)

Step 4: Terminate (Degree of 2 is 0, which is < Degree of x (1))

So, for $A(x) = x^3 + 5x^2 + 6x + 2$ and $B(x) = x + 2$: The Quotient is $H(x) = x^2 + 3x$. The Remainder is $R(x) = 2$.

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

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

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. ✅