Follow the steps:

Bring down 2.

2 × 2 = 4

3 + 4 = 7

7 × 2 = 14

−5 + 14 = 9

9 × 2 = 18

7 + 18 = 25

Step 4: Interpret

The quotient coefficients are:

2, 7, 9

The quotient is:

Q(x) = 2x² + 7x + 9

The remainder is:

R = 25

Therefore:

(2x³ + 3x² − 5x + 7)/(x − 2) = 2x² + 7x + 9 + 25/(x − 2)

for x ≠ 2.

Verification

Multiply the divisor and quotient:

(x − 2)(2x² + 7x + 9)

= 2x³ + 7x² + 9x − 4x² − 14x − 18

= 2x³ + 3x² − 5x − 18

Add the remainder:

2x³ + 3x² − 5x − 18 + 25

= 2x³ + 3x² − 5x + 7

The original dividend is recovered.

Synthetic Division with Missing Terms

Missing terms must be represented by zero coefficients.

Worked Example

Divide:

(x⁴ − 3x² + 2x − 8) ÷ (x + 2)

The dividend has no x³ term.

Rewrite:

x⁴ + 0x³ − 3x² + 2x − 8

The divisor is x + 2, so:

c = −2

Set up:

The quotient coefficients are:

1, −2, 1, 0

Therefore:

Q(x) = x³ − 2x² + x

The remainder is:

R = −8

Final Result

(x⁴ − 3x² + 2x − 8)/(x + 2) = x³ − 2x² + x − 8/(x + 2)

where x ≠ −2.

Important Reminder

If a power is missing, include a zero in its position.

Otherwise, every coefficient after the missing term may be interpreted incorrectly.

Comparing Long Division and Synthetic Division

When Should You Use Each Method?

Use long division when:

• The divisor has degree 2 or higher.

• You want a general method that works with any polynomial divisor.

• You need to see the subtraction and multiplication steps explicitly.

Use synthetic division when:

• The divisor is x − c.

• You want a quicker calculation.

• You need to test whether x − c is a factor.

For divisors such as 2x − 3, the standard synthetic method requires an adjustment. Long division is a reliable choice.

Dividing by a Linear Expression with a Leading Coefficient

Consider:

(2x² + 7x + 3) ÷ (2x + 1)

We can use long division.

Step 1

2x² ÷ 2x = x

Multiply:

x(2x + 1) = 2x² + x

Subtract:

(2x² + 7x + 3) − (2x² + x)

= 6x + 3

Step 2

6x ÷ 2x = 3

Multiply:

3(2x + 1) = 6x + 3

Subtract:

0

Therefore:

Quotient = x + 3

Remainder = 0

Using a Modified Synthetic Approach

The divisor can also be written as:

2x + 1 = 2(x + 1/2)

We can synthetically divide by x + 1/2, using c = −1/2.

This produces:

2x + 6

Because the original divisor is twice x + 1/2, divide this quotient by 2:

(2x + 6)/2 = x + 3

The remainder remains zero.

This method works, but the adjustment must not be forgotten.

The Remainder Theorem

The Remainder Theorem connects polynomial division with function evaluation.

It states:

When a polynomial P(x) is divided by x − c, the remainder is:

R = P(c)

This means we can find the remainder without performing long division.

Worked Example

Find the remainder when:

P(x) = x³ − 4x + 7

is divided by:

x − 2

The divisor has c = 2.

Evaluate:

P(2) = 2³ − 4(2) + 7

= 8 − 8 + 7

Remainder = 7

Another Example

Find the remainder when:

P(x) = 2x³ + 5x² − x + 4

is divided by:

x + 1

Here:

c = −1

Evaluate:

P(−1) = 2(−1)³ + 5(−1)² − (−1) + 4

= −2 + 5 + 1 + 4

Remainder = 8

Why Does This Work?

Polynomial division gives:

P(x) = (x − c)Q(x) + R

Substitute x = c:

P(c) = (c − c)Q(c) + R

P(c) = 0 + R

Therefore:

P(c) = R

The Factor Theorem

The Factor Theorem is closely related to the Remainder Theorem.

It states:

x − c is a factor of P(x) if and only if P(c) = 0.

If the remainder is zero, the division is exact.

Worked Example

Determine whether x − 3 is a factor of:

P(x) = x² − 5x + 6

Evaluate:

P(3) = 3² − 5(3) + 6

= 9 − 15 + 6

= 0

Because the remainder is zero:

x − 3 is a factor.

We can verify by factoring:

x² − 5x + 6 = (x − 2)(x − 3)

Another Example

Is x + 2 a factor of:

P(x) = x³ + 3x² − 4x + 5?

Evaluate P(−2):

P(−2) = (−2)³ + 3(−2)² − 4(−2) + 5

= −8 + 12 + 8 + 5

= 17

Because P(−2) ≠ 0:

x + 2 is not a factor.

Using Synthetic Division to Factor Polynomials

Synthetic division can help factor higher-degree polynomials after one factor has been identified.

Worked Example

Factor:

P(x) = x³ − 6x² + 11x − 6

Test x = 1:

P(1) = 1 − 6 + 11 − 6 = 0

Therefore, x − 1 is a factor.

Use synthetic division:

The quotient is:

x² − 5x + 6

Factor the quotient:

x² − 5x + 6 = (x − 2)(x − 3)

Therefore:

P(x) = (x − 1)(x − 2)(x − 3)

Interpretation

The polynomial has three zeros:

x = 1

x = 2

x = 3

Synthetic division helps us move from a cubic polynomial to a simpler quadratic polynomial.

Interpreting Quotients and Remainders

The quotient tells us how many times the divisor fits into the dividend in an algebraic sense.

The remainder describes the part left over after the polynomial quotient has been removed.

For example:

x² + 5x + 7 = (x + 2)(x + 3) + 1

This tells us:

• Dividend: x² + 5x + 7

• Divisor: x + 2

• Quotient: x + 3

• Remainder: 1

The quotient is not the complete rational expression when the remainder is nonzero.

We must write:

(x² + 5x + 7)/(x + 2) = x + 3 + 1/(x + 2)

Degree of the Remainder

The remainder must have a degree lower than the divisor.

Examples:

• Dividing by a linear polynomial produces a constant remainder or zero.

• Dividing by a quadratic polynomial can produce a linear remainder, a constant remainder, or zero.

• Dividing by a cubic polynomial can produce a quadratic remainder or one of lower degree.

This condition tells us when polynomial division is complete.

Polynomial Division and Rational Expressions

Polynomial division is especially useful when the numerator of a rational expression has degree greater than or equal to the denominator.

Consider:

(x² + 3x + 5)/(x + 1)

Long division gives:

Quotient = x + 2

Remainder = 3

Therefore:

(x² + 3x + 5)/(x + 1) = x + 2 + 3/(x + 1)

where x ≠ −1.

This form separates the polynomial part from the remaining fraction.

It is useful in advanced algebra, calculus, and the study of rational functions.

Understanding Polynomial Division Through Area

Multiplication and division are inverse operations.

For example, if a rectangle has area:

A = x² + 5x + 6

and one side has length:

x + 2

we can determine the other side by division:

(x² + 5x + 6) ÷ (x + 2)

The quotient is:

x + 3

Therefore, the rectangle's dimensions are:

x + 2

and:

x + 3

Polynomial division can therefore be understood as finding an unknown factor in a product.

Verifying Polynomial Division

Every polynomial division result can be checked using:

Dividend = Divisor × Quotient + Remainder

This is called the division algorithm for polynomials.

Worked Example

Verify the division:

(2x³ + 5x² − x + 4) ÷ (x + 3)

Suppose the quotient is:

Q(x) = 2x² − x + 2

and the remainder is:

R = −2

Check by multiplying:

(x + 3)(2x² − x + 2)

Distribute x:

2x³ − x² + 2x

Distribute 3:

6x² − 3x + 6

Combine:

2x³ + 5x² − x + 6

Add the remainder:

2x³ + 5x² − x + 6 − 2

= 2x³ + 5x² − x + 4

The original dividend is recovered.

The division is correct.

A Second Check Using the Remainder Theorem

Because the divisor is x + 3, evaluate the dividend at x = −3.

P(−3) = 2(−3)³ + 5(−3)² − (−3) + 4

= −54 + 45 + 3 + 4

= −2

This agrees with the remainder.

Using both checks can help detect calculation errors.

Real-World Application: Finding an Unknown Dimension

Suppose a rectangular storage area has an area of:

A = 2x² + 9x + 10

One side has length:

x + 2

Find the other side.

Use polynomial division:

(2x² + 9x + 10) ÷ (x + 2)

Step 1

2x² ÷ x = 2x

Multiply:

2x(x + 2) = 2x² + 4x

Subtract:

5x + 10

Step 2

5x ÷ x = 5

Multiply:

5(x + 2) = 5x + 10

Subtract:

0

Therefore:

Other side = 2x + 5

Verification

(x + 2)(2x + 5)

= 2x² + 9x + 10

The area expression is recovered.

If x = 3 metres:

First side = 5 m

Second side = 11 m

Area = 5 × 11

Area = 55 m²

Real-World Application: Production and Packaging

Suppose a manufacturer models the total material required for a production process using:

M(x) = 3x² + 14x + 8

The material required per package is represented by:

D(x) = x + 4

To find the polynomial quotient, divide:

(3x² + 14x + 8) ÷ (x + 4)

Long Division

3x² ÷ x = 3x

Multiply:

3x(x + 4) = 3x² + 12x

Subtract:

2x + 8

Divide:

2x ÷ x = 2

Multiply:

2(x + 4) = 2x + 8

Subtract:

0

Therefore:

Q(x) = 3x + 2

Interpretation

Within this algebraic model, the total material expression is exactly the product:

M(x) = (x + 4)(3x + 2)

The quotient represents the corresponding number of packages algebraically, provided the model's units and assumptions support that interpretation.

Common Mistakes

Subtracting exponents incorrectly

Incorrect:

x⁶ ÷ x² = x³

Correct:

x⁶ ÷ x² = x⁴

Dividing only the first term by a monomial

Incorrect:

(6x² + 9x)/3x = 2x + 9x

Correct:

(6x² + 9x)/3x = 2x + 3

Forgetting missing terms

For synthetic division of x³ + 4x − 7, use coefficients:

1, 0, 4, −7

Using the wrong sign in synthetic division

For x + 3, use c = −3, not 3.

Forgetting to subtract the entire polynomial during long division

Every term in the polynomial being subtracted must change sign.

Stopping long division too early

Continue until the remainder has a lower degree than the divisor.

Treating the remainder as part of the quotient

If the quotient is x + 2 and the remainder is 3, the rational expression is:

x + 2 + 3/(x + 1)

not x + 5.

Ignoring the remainder

A nonzero remainder must be included in the final division result.

Forgetting domain restrictions

The divisor cannot equal zero.

For example:

(x² − 4)/(x − 2) = x + 2

only when x ≠ 2.

Using basic synthetic division with an unsuitable divisor

The standard shortcut is designed for x − c. For other divisors, use long division or an appropriately modified method.

Failing to verify

Always check:

Divisor × Quotient + Remainder = Dividend

Key Terms

• Polynomial Division: Process of dividing one polynomial by another.

• Dividend: Polynomial being divided.

• Divisor: Polynomial by which another polynomial is divided.

• Quotient: Polynomial result of the division algorithm.

• Remainder: Polynomial left after division, with degree lower than the divisor.

• Monomial: Polynomial containing one term.

• Leading Term: Term with the highest power in a polynomial.

• Leading Coefficient: Coefficient of the leading term.

• Degree: Highest exponent in a nonzero polynomial in one variable.

• Quotient of Powers Rule: Rule for subtracting exponents when dividing powers with the same nonzero base.

• Long Division: Systematic polynomial division using repeated multiplication and subtraction.

• Synthetic Division: Shortcut using coefficients when dividing by a suitable linear polynomial.

• Zero Coefficient: Coefficient used to represent a missing power.

• Division Algorithm: Relationship P(x) = D(x)Q(x) + R(x).

• Remainder Theorem: When dividing P(x) by x − c, the remainder equals P(c).

• Factor Theorem: x − c is a factor of P(x) exactly when P(c) = 0.

• Exact Division: Division producing a zero remainder.

• Rational Expression: Quotient of two polynomials with a nonzero denominator.

• Domain Restriction: Value excluded because it makes the original denominator zero.

• Verification: Checking a division result using multiplication and addition.

• Zero of a Polynomial: Value of x that makes the polynomial equal zero.

Key Takeaways

• Polynomial division extends ordinary numerical division.

• When dividing powers with the same base, subtract their exponents.

• Dividing a polynomial by a monomial requires dividing every term.

• Polynomial division produces a quotient and a remainder.

• The degree of a nonzero remainder must be lower than the degree of the divisor.

• Long division works with any nonzero polynomial divisor.

• Synthetic division is a shortcut for divisors of the form x − c.

• Missing powers must be represented by zero coefficients.

• For a divisor x + a, synthetic division uses c = −a.

• The Remainder Theorem allows us to calculate remainders by substitution.

• The Factor Theorem helps identify factors of polynomials.

• A zero remainder means the divisor is a factor of the dividend.

• Polynomial division can help factor higher-degree polynomials.

• A nonzero remainder must be included in the final rational expression.

• Every division result can be verified using the division algorithm.

Check Your Understanding

Dividing by Monomials

Simplify each expression.

1. 12x⁴ ÷ 3x²

2. −20a⁵ ÷ 5a²

3. (15x³ + 10x²)/5x

4. (18x⁴ − 12x³ + 6x²)/6x²

Polynomial Long Division

Find the quotient and remainder.

5. (x² + 7x + 12) ÷ (x + 3)

6. (x² + 5x + 9) ÷ (x + 2)

7. (2x² + 9x + 7) ÷ (x + 1)

8. (x³ + 3x² + 2x) ÷ (x + 1)

9. (x³ + 2x² + 5x + 4) ÷ (x² + 1)

10. (2x³ + 5x² + 4x + 3) ÷ (x² + 2x + 1)

Synthetic Division

Use synthetic division to find the quotient and remainder.

11. (x² − 5x + 6) ÷ (x − 2)

12. (x³ − 4x² + x + 6) ÷ (x − 3)

13. (2x³ + 3x² − 5x + 7) ÷ (x + 1)

14. (x⁴ − 5x² + 4) ÷ (x − 2)

15. (3x³ − 2x² + 5x − 1) ÷ (x + 2)

Remainders and Verification

16. Find the remainder when x² + 4x + 7 is divided by x − 2.

17. Determine whether x − 3 is a factor of x² − 7x + 12.

18. Determine whether x + 2 is a factor of x³ + 2x² − 4x − 8.

19. Explain how to verify a polynomial division result.

20. Explain why the remainder must have a lower degree than the divisor.

Answer Key

For Questions 1–4, retain the domain restrictions imposed by the original divisors.

Extended Challenge

Dividing by Monomials

1. (24x⁶ − 18x⁴ + 12x²)/6x²

2. (30a⁵b³ − 20a³b² + 10a²b)/5a²b

3. (−16x⁵ + 24x³ − 8x²)/(−8x²)

4. (12x⁴ + 9x² + 3)/3x

5. Explain why dividing a polynomial by a monomial does not always produce another polynomial.

Polynomial Long Division

Find the quotient and remainder.

6. (x³ + 5x² + 8x + 4) ÷ (x + 1)

7. (2x³ − 3x² + 4x − 5) ÷ (x − 2)

8. (x⁴ − 3x² + 2x − 8) ÷ (x + 2)

9. (3x⁴ + 2x³ − 5x² + x + 7) ÷ (x² + 1)

10. (2x⁴ − x³ + 3x² − 4x + 5) ÷ (x² − x + 2)

Synthetic Division

11. (x³ − 7x + 6) ÷ (x − 1)

12. (2x⁴ − 3x³ + 5x − 8) ÷ (x + 2)

13. (3x³ + 4x² − 7x + 2) ÷ (x − 1)

14. (x⁵ − 4x³ + 3x² − 2) ÷ (x + 1)

15. (4x⁴ − 5x³ + 2x² − 7x + 9) ÷ (x − 3)

Remainder and Factor Theorems

16. Find the remainder when 2x³ − 5x² + 4x − 7 is divided by x − 2.

17. Determine whether x + 3 is a factor of x³ + 2x² − 5x + 6.

18. Determine whether x − 4 is a factor of x³ − 6x² + 8x.

19. Find a value of k so that x − 2 is a factor of x³ + kx² − 4x + 8.

20. Explain how synthetic division and the Factor Theorem can work together to factor a cubic polynomial.

Advanced Applications

21. Divide x⁴ − 1 by x² − 1.

22. Divide 2x⁵ + 3x³ − x + 4 by x² + x + 1.

23. Find the quotient and remainder when x⁵ − 3x⁴ + 2x² − 7 is divided by x − 3.

24. Use synthetic division to factor x³ − 4x² + x + 6 completely.

25. Construct a cubic polynomial that gives quotient x² + 2x − 3 and remainder 5 when divided by x + 2.

Final Challenge: Investigating Polynomial Division

A polynomial is defined by:

P(x) = x³ − 4x² − x + 4

Complete the investigation.

1. Identify the degree of P(x).

2. Evaluate P(1).

3. Use the Remainder Theorem to determine the remainder when P(x) is divided by x − 1.

4. Determine whether x − 1 is a factor.

5. Divide P(x) by x − 1 using long division.

6. Repeat the division using synthetic division.

7. Compare the quotients obtained by both methods.

8. Verify the division using multiplication.

9. Factor the resulting quadratic quotient.

10. Write P(x) as a product of linear factors.

11. Identify all zeros of P(x).

12. Explain how these zeros relate to the x-intercepts of its graph.

13. Determine the remainder when P(x) is divided by x + 2.

14. Verify that remainder using substitution.

15. Explain why synthetic division is efficient for linear divisors.

16. Describe a situation in which long division would be preferable.

17. Explain the meaning of a zero remainder.

18. Explain why a nonzero remainder prevents the divisor from being a factor.

19. Write a general statement describing the relationship between polynomial division and factoring.

20. Summarise how long division, synthetic division, and the Remainder Theorem are connected.

Further Investigation: Creating Your Own Polynomial

Choose three different integer values for a, b, and c.

Construct:

P(x) = (x − a)(x − b)(x − c)

Expand the expression to obtain a cubic polynomial.

Then:

• Divide P(x) by x − a using synthetic division.

• Divide P(x) by x − b using long division.

• Use the Remainder Theorem to determine the remainder when dividing by x − c.

• Verify each result.

• Explain why all three divisions produce a zero remainder.

• Change one coefficient of your expanded polynomial and investigate how the remainders change.

This investigation demonstrates how polynomial multiplication, factoring, and division are closely related operations.