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C. Polynomial and Rational Functions · Polynomial Functions and Equations

Dividing Polynomials: Long Division and Synthetic Division

C3.1About 75 minutes
Learning goals
  • I can divide a polynomial by a linear or quadratic polynomial using long division.
  • I can use synthetic division to divide by x − b, including when some powers are missing.
  • I can write the division statement P(x) = D(x)Q(x) + R(x) and check it by expanding.
  • I can divide by a linear divisor such as 2x − 1 whose leading coefficient is not 1.

Words to know

dividend
The polynomial being divided, P(x).
divisor
The polynomial you divide by, D(x).
quotient
The result of the division, Q(x).
remainder
What is left over, R(x). Its degree is less than the degree of the divisor (so dividing by a linear divisor leaves a constant remainder).
division statement
P(x) = D(x) · Q(x) + R(x).
synthetic division
A short way to divide by x − b using only the coefficients.
placeholder
A 0 written for a missing power of x (for example 3x⁴ + 0x³ − 5x² ...).

Why divide polynomials?

In Grade 10 you factored quadratics by spotting patterns. For a cubic or quartic, patterns are rare. The main tool is division: if you know one factor, dividing by it leaves a smaller polynomial that is easier to factor. Division also tells you the remainder, which leads to the remainder and factor theorems in the next two lessons.

Dividing polynomials works just like the long division of numbers you learned in elementary school: divide, multiply, subtract, bring down, repeat. The result is written as a division statement.

437=12×36+5P(x)=D(x)·Q(x)+R(x)dividenddivisorquotientremainderdegree of R(x) < degree of D(x)dividing numbers and dividing polynomials follow the same pattern
A division statement for numbers and for polynomials. For polynomials, you stop when the remainder's degree is less than the divisor's degree.
Worked example

Example 1: Long division by x + 3

Divide 2x³ + 3x² − 5x + 6 by x + 3. Write the division statement.

  1. Divide leading terms: 2x³ ÷ x = 2x². Multiply: 2x²(x + 3) = 2x³ + 6x². Subtract: 3x² − 6x² = −3x². Bring down −5x: −3x² − 5x.
  2. Divide: −3x² ÷ x = −3x. Multiply: −3x(x + 3) = −3x² − 9x. Subtract: −5x − (−9x) = 4x. Bring down 6: 4x + 6.
  3. Divide: 4x ÷ x = 4. Multiply: 4(x + 3) = 4x + 12. Subtract: 6 − 12 = −6. The remainder −6 has degree 0, less than 1, so stop.
  4. Check by expanding: (x + 3)(2x² − 3x + 4) = 2x³ − 3x² + 4x + 6x² − 9x + 12 = 2x³ + 3x² − 5x + 12. Add −6: 2x³ + 3x² − 5x + 6 ✓.

Quotient 2x² − 3x + 4, remainder −6: 2x³ + 3x² − 5x + 6 = (x + 3)(2x² − 3x + 4) − 6. (You can also write (2x³ + 3x² − 5x + 6)/(x + 3) = 2x² − 3x + 4 − 6/(x + 3), x ≠ −3.)

2x²− 3x+ 4x + 32x³+ 3x²− 5x+ 6−(2x³+ 6x²)−3x²− 5x−(−3x²− 9x)4x+ 6−(4x+ 12)−6← remainder1. divide leading terms2. multiply3. subtract4. bring down; repeat
The long division from Example 1, with terms lined up by power. Watch the signs: you subtract the whole bracket each time.

Synthetic division: a shortcut for x − b

When the divisor is x − b, you can skip writing the x's. Write b in a box and the coefficients of the dividend in a row (with 0 for any missing power). Then:

  1. Bring down the first coefficient.
  2. Multiply it by b and write the result under the next coefficient.
  3. Add that column.
  4. Repeat multiply-and-add to the end.

The last number is the remainder. The other numbers are the coefficients of the quotient, which has degree one less than the dividend.

Careful with the sign: to divide by x + 3, use b = −3, because x + 3 = x − (−3).

−3223−6−3−5946−12−6−6× −3quotient 2x² − 3x + 4remainder −6
Synthetic division for (2x³ + 3x² − 5x + 6) ÷ (x + 3), with b = −3. Bring down 2; then multiply by −3 and add, column by column: 2, −3, 4, and remainder −6.
Worked example

Example 2: Synthetic division with a missing power

Divide 3x⁴ − 5x² + x − 2 by x − 2.

  1. b = 2. The x³ term is missing, so the coefficients are 3, 0, −5, 1, −2.
  2. Bring down 3. 3 × 2 = 6; 0 + 6 = 6. 6 × 2 = 12; −5 + 12 = 7. 7 × 2 = 14; 1 + 14 = 15. 15 × 2 = 30; −2 + 30 = 28.
  3. Bottom row: 3, 6, 7, 15 | 28. The quotient has degree 3.

3x⁴ − 5x² + x − 2 = (x − 2)(3x³ + 6x² + 7x + 15) + 28

Worked example

Example 3: Dividing by 2x − 1

Divide 4x³ − 2x² + 6x − 1 by 2x − 1.

  1. Long division: 4x³ ÷ 2x = 2x². 2x²(2x − 1) = 4x³ − 2x². Subtract: 0x². Bring down: 6x − 1.
  2. 0x² needs a quotient term of 0x. Then 6x ÷ 2x = 3. 3(2x − 1) = 6x − 3. Subtract: −1 − (−3) = 2.
  3. Or synthetic: 2x − 1 = 2(x − ½). Divide by x − ½: coefficients 4, −2, 6, −1 give 4, 0, 6 | 2. That quotient 4x² + 6 is for x − ½; divide it by 2 to get the quotient for 2x − 1: 2x² + 3. The remainder stays 2.
  4. Check: (2x − 1)(2x² + 3) + 2 = 4x³ + 6x − 2x² − 3 + 2 = 4x³ − 2x² + 6x − 1 ✓.

4x³ − 2x² + 6x − 1 = (2x − 1)(2x² + 3) + 2

Worked example

Example 4: Dividing by a quadratic

Divide x⁴ + 2x³ − 3x² + x − 5 by x² + x − 1.

  1. x⁴ ÷ x² = x². x²(x² + x − 1) = x⁴ + x³ − x². Subtract: x³ − 2x² + x − 5.
  2. x³ ÷ x² = x. x(x² + x − 1) = x³ + x² − x. Subtract: −3x² + 2x − 5.
  3. −3x² ÷ x² = −3. −3(x² + x − 1) = −3x² − 3x + 3. Subtract: 5x − 8.
  4. 5x − 8 has degree 1, less than 2, so stop. (Synthetic division only works for linear divisors x − b; use long division here.)

Quotient x² + x − 3, remainder 5x − 8: x⁴ + 2x³ − 3x² + x − 5 = (x² + x − 1)(x² + x − 3) + 5x − 8

Technology check

A computer algebra system (CAS) such as the free GeoGebra CAS can divide polynomials for you: for example, Division(2x^3 + 3x^2 - 5x + 6, x + 3) returns the quotient and remainder. Use it to check your work, but practise by hand: you will need these steps in every lesson that follows.

Just for fun

The shortcut behind synthetic division is often called Horner's method, after William Horner, who published it in 1819. The Chinese mathematician Qin Jiushao described the same idea in 1247, about 570 years earlier.

Hands-on activity

Polynomial division is number division in disguise

You need: paper, a pencil and a calculator.

  1. Divide 1331 by 11 by hand (long division).
  2. Divide x³ + 3x² + 3x + 1 by x + 1 (long or synthetic division).
  3. Substitute x = 10 into the divisor, the dividend and your quotient. What do you notice?
  4. Now divide 2x³ − 7x² + 0x + 5 by x − 3, and then substitute x = 10. Compare with 1305 ÷ 7 on your calculator.

What to look for: 1331 ÷ 11 = 121, and (x³ + 3x² + 3x + 1) ÷ (x + 1) = x² + 2x + 1, which is 121 when x = 10. For step 4, the quotient is 2x² − x − 3 with remainder −4. At x = 10: 7 × 187 − 4 = 1309 − 4 = 1305 ✓. With numbers we prefer a remainder between 0 and 6 (1305 = 7 × 186 + 3), but polynomial remainders can be negative: the two statements are both true.

Your turn

1

Divide x³ + 3x² + 3x + 1 by x + 1.

2

Divide 2x³ − 7x² + 5 by x − 3.

3

Divide x³ − 8 by x − 2.

4

Divide 4x³ − 3x + 7 by 2x + 1.

5

Write the division statement for 3x³ − x² + 4x − 2 divided by x + 1.

6

Divide x⁴ − 16 by x + 2.

7

Divide 2x³ + x² − 13x + 6 by x² + 2x − 3.

More practice: warm-up

1

A degree-4 polynomial is divided by x − 5. What is the degree of the quotient?

2

When you divide by a linear polynomial, what is the greatest possible degree of the remainder?

3

What coefficients do you write for synthetic division of x³ + 4x − 1?

4

Write a division statement for 17 ÷ 5.

5

Which value of b do you use in synthetic division by x + 4?

More practice: on your own

1

Divide x⁴ − 4x³ + 5x + 3 by x² + 1.

2

Divide 2x³ − 3x² + 4 by x − 1.

3

P(x) divided by x − 2 gives quotient x² + 3x − 1 and remainder 5. Find P(x).

4

Divide 2x³ − x² − 7x + 10 by 2x − 3.

5

Use division to decide whether x + 3 divides x³ + 27 evenly.

Challenge

1

A box has volume V(x) = x³ + 6x² + 11x + 6 (cm³) and height x + 1. Find the area of its base, then find possible length and width.

2

Find k so that x³ + kx² − 4x + 1 divided by x − 1 has remainder 0.

3

When P(x) is divided by x² − 1, the quotient is x + 2 and the remainder is 3x − 1. Find P(x).

Check your understanding

Quiz

1.

What is the remainder when x³ − 2x² + 4 is divided by x − 3?

2.

What is the quotient when 2x³ + 5x² − x − 6 is divided by x + 2?

3.

Which value do you place in the box for synthetic division by x + 5?

4.

Which division statement is correct?

5.

A polynomial of degree 5 is divided by a polynomial of degree 2. What is the greatest possible degree of the remainder?

6.

What is the remainder when 4x³ − 3x + 7 is divided by 2x + 1?

For parents and teachers

Big idea (C3.1): polynomial division follows the same steps as long division of numbers (divide, multiply, subtract, bring down) and ends with a division statement P(x) = D(x)Q(x) + R(x). Synthetic division is a fast shortcut when dividing by x − b.

How to help: Ask your teen to check each answer by multiplying the divisor by the quotient and adding the remainder; it should give back the original polynomial. This habit catches almost every error.

Common difficulties:

  • Forgetting a 0 placeholder for a missing power.
  • Using b = 3 instead of b = −3 for the divisor x + 3.
  • Subtracting only the first term of a bracket in long division (subtract every term).
  • Dividing by 2x − 1 with synthetic division and forgetting to divide the quotient by 2.