Dividing by a monomial
Share 12x + 8 equally among 4 friends: each gets (12x + 8) ÷ 4 = 3x + 2. Divide every term separately.
To divide a polynomial by a monomial, divide each term by the monomial and write the results.
In the figure: Share 12x + 8 into 4 equal parts: divide each term separately ⇒ each part is 3x + 2.
Formula: (term1 + term2) ÷ divisor = term1 ÷ divisor + term2 ÷ divisor
In words: Divide the numbers, and subtract the powers of the same letter (x³ ÷ x = x²).
- Easy: (12x³ + 8x²) ÷ (4x) = 3x² + 2x
- Medium: (6m²n + 9mn²) ÷ (3mn) = 2m + 3n
- Exam-type: (20a³b − 15a²b² + 10ab³) ÷ (5ab) = 4a² − 3ab + 2b²
Check: quotient × divisor should give back the dividend.
Do not divide only the first term; divide every term.
When the divisor is a monomial, divide the numbers and subtract the powers of the same letter. Memory clue: In division first the numbers, then the letters: "divide the numbers, subtract the powers".
Spot the difference
| Point | Dividing by a monomial | Dividing by a binomial |
|---|---|---|
| Method | divide each term separately | long division |
| Remainder | usually 0 | 0 or non-zero |
| Example | (12x² + 8x) ÷ 4x = 3x + 2 | (x² + 7x + 10) ÷ (x + 2) = x + 5 |
| Check | quotient × divisor | divisor × quotient + remainder |
Practice questions
- 1(10x² + 15x) ÷ 5x
- 2(18a³ − 12a²) ÷ 6a
- 3(14m²n + 21mn²) ÷ 7mn
- 4(24x³ − 16x² + 8x) ÷ 8x
- 5(9p³q² − 6p²q³) ÷ 3p²q²
- 6(35x⁴ + 28x³ − 14x²) ÷ 7x²
- 7A rectangle has area 24x² + 36x and length 6x. Find its breadth.
- 8(20a²b² − 15ab³ + 5ab) ÷ 5ab and check.
Show answers (try them yourself first)
- 12x + 3
- 23a² − 2a
- 32m + 3n
- 43x² − 2x + 1
- 53p − 2q
- 65x² + 4x − 2
- 7Breadth = (24x² + 36x) ÷ 6x = 4x + 6.
- 8Quotient = 4ab − 3b² + 1; check: 5ab × (4ab − 3b² + 1) = 20a²b² − 15ab³ + 5ab ✔.
Dividing by a binomial (long division)
47 ÷ 5 gives quotient 9 and remainder 2, written 47 = 5 × 9 + 2. Polynomial division works exactly the same way, with powers of x instead of digits.
Dividend = divisor × quotient + remainder. Stop once the remainder is of lower degree than the divisor.
In the figure: (x² + 7x + 10) ÷ (x + 2): x² ÷ x = x; subtracting gives 5x + 10; 5x ÷ x = 5; subtracting gives 0 ⇒ quotient x + 5.
Formula: Dividend = Divisor × Quotient + Remainder
In words: Steps: 1) Write both in descending powers. 2) Divide the first term of the dividend by the first term of the divisor — that is the next term of the quotient. 3) Multiply it by the divisor and write it below. 4) Subtract and bring down the next term. 5) Repeat until the remainder is smaller.
Dividing by a binomial (long division)
In the app you can move this figure with your finger.
Open in the app- Easy: (x² + 7x + 10) ÷ (x + 2) = x + 5 (remainder 0)
- Medium: x² + 5x + 9 = (x + 2)(x + 3) + 3 (quotient x + 3, remainder 3)
- Exam-type: (2x³ + 3x² − 5x − 6) ÷ (x + 2) = 2x² − x − 3 (remainder 0)
Steps (easy): x² ÷ x = x ; x(x + 2) = x² + 2x ; subtract gives 5x + 10 ; 5x ÷ x = 5 ; 5(x + 2) = 5x + 10 ; subtract gives 0.
Steps (exam-type): 2x³ ÷ x = 2x² ; 2x²(x + 2) = 2x³ + 4x² ; subtract: −x² − 5x − 6 ; −x² ÷ x = −x ; −x(x + 2) = −x² − 2x ; subtract: −3x − 6 ; −3(x + 2) = −3x − 6 ; remainder 0.
Missing terms: x³ − 8 has no x² or x term. Write them as 0x² and 0x: x³ + 0x² + 0x − 8. Then (x³ − 8) ÷ (x − 2) = x² + 2x + 4.
Dividend = divisor × quotient + remainder — always use it to check.
Do not forget to change signs while subtracting: −(x² + 2x) = −x² − 2x.
If the remainder is 0 the divisor is a factor of the dividend. Memory clue: It works like long division: dividend, divisor, quotient and remainder; "quotient × divisor + remainder = dividend".
Spot the difference
| Point | Numbers 47 ÷ 5 | Polynomials (x² + 5x + 9) ÷ (x + 2) |
|---|---|---|
| Quotient | 9 | x + 3 |
| Remainder | 2 | 3 |
| Check | 5 × 9 + 2 = 47 | (x + 2)(x + 3) + 3 = x² + 5x + 9 |
Practice questions
- 1(15a³ − 10a²) ÷ (5a)
- 2(8m²n + 12mn²) ÷ (4mn)
- 3(x² + 9x + 20) ÷ (x + 4)
- 4(x² + x − 12) ÷ (x − 3)
- 5(x² + 6x + 10) ÷ (x + 3): quotient and remainder?
- 6(x³ + 27) ÷ (x + 3)
- 7(6x² + 7x − 3) ÷ (2x + 3)
- 8What must be added to x² + 4x + 1 so that it is exactly divisible by (x + 2)?
Show answers (try them yourself first)
- 1(15a³ − 10a²) ÷ (5a) = 3a² − 2a
- 2(8m²n + 12mn²) ÷ (4mn) = 2m + 3n
- 3(x² + 9x + 20) ÷ (x + 4) = x + 5
- 4(x² + x − 12) ÷ (x − 3) = x + 4
- 5x² + 6x + 10 = (x + 3)(x + 3) + 1 (quotient x + 3, remainder 1)
- 6(x³ + 27) ÷ (x + 3) = x² − 3x + 9
- 7(6x² + 7x − 3) ÷ (2x + 3) = 3x − 1
- 8x² + 4x + 1 = (x + 2)(x + 2) − 3 so add 3: x² + 4x + 4 = (x + 2)², which divides exactly.