Dividing Polynomials by Binomials With Example
Dividing polynomials by binomials is an important concept in algebra. It helps simplify algebraic expressions, solve polynomial equations, and understand how polynomial functions behave. This concept is widely used in higher mathematics, including calculus and algebraic modeling.
When we divide a polynomial by a binomial, we are finding how many times the binomial fits into the polynomial, just like regular long division with numbers.
For example:
(x² + 5x + 6) ÷ (x + 2)
To solve problems like this, we use methods such as Polynomial Long Division or Synthetic Division.
What Does Dividing Polynomials Mean?
Dividing polynomials means expressing one polynomial (the dividend) as:
Dividend = (Divisor × Quotient) + Remainder
Just like number division:
15 ÷ 4 = 3 remainder 3
Similarly, in polynomials:
x² + 5x + 6 = (x + 2)(x + 3) + 0
If the remainder is 0, the binomial is a factor of the polynomial.
Polynomial Long Division Method
Polynomial long division is similar to regular long division with numbers.
Steps for Polynomial Long Division
- Arrange both polynomials in descending order of powers.
- Divide the first term of the dividend by the first term of the divisor.
- Multiply the entire divisor by that result.
- Subtract.
- Repeat until no terms are left.
Example
Divide:
(x² + 5x + 6) ÷ (x + 2)
Step 1: Divide the first terms
x² ÷ x = x
Step 2: Multiply
x(x + 2) = x² + 2x
Step 3: Subtract
(x² + 5x + 6)
− (x² + 2x)
= 3x + 6
Step 4: Divide again
3x ÷ x = 3
Step 5: Multiply
3(x + 2) = 3x + 6
Step 6: Subtract
(3x + 6) − (3x + 6) = 0
Final Answer:
Quotient = x + 3
Remainder = 0
So,
(x² + 5x + 6) ÷ (x + 2) = x + 3
Example (With Remainder)
Divide:
(2x² + 3x + 1) ÷ (x + 1)
Step 1: Divide first terms
2x² ÷ x = 2x
Step 2: Multiply
2x(x + 1) = 2x² + 2x
Step 3: Subtract
(2x² + 3x + 1)
− (2x² + 2x)
= x + 1
Step 4: Divide again
x ÷ x = 1
Step 5: Multiply
1(x + 1) = x + 1
Step 6: Subtract
(x + 1) − (x + 1) = 0
Final Answer:
Quotient = 2x + 1
Remainder = 0
Synthetic Division (Shortcut Method)
Synthetic division is a faster method used when dividing by a binomial of the form:
x − a
For example:
(x² + 4x + 3) ÷ (x − 1)
Instead of writing the full long division, we use the number 1 (since x − 1 = 0 gives x = 1).
Steps for Synthetic Division
- Write the coefficients of the polynomial.
- Use the zero of the divisor.
- Perform simple multiplication and addition.
Example
Divide:
(x² + 4x + 3) ÷ (x − 1)
Coefficients: 1 4 3
Zero: 1
Bring down 1
Multiply 1 × 1 = 1
Add to 4 → 5
Multiply 5 × 1 = 5
Add to 3 → 8
Result:
Quotient = x + 5
Remainder = 8
So,
(x² + 4x + 3) ÷ (x − 1) = x + 5 with remainder 8
Common Mistakes to Avoid
1. Not Writing in Descending Order
Always arrange the polynomial in descending powers of x.
2. Forgetting Missing Terms
If a term is missing, add 0 as a placeholder.
Example:
x³ + 2x + 1
Write as:
x³ + 0x² + 2x + 1
3. Sign Errors While Subtracting
Be careful when subtracting expressions in long division.
