Polynomial division | Year 12

01 / LEARN

The idea, step by step

The division identity

For a non-zero divisor $D (x)$ , polynomial division writes $P (x) = D (x) Q (x) + R (x)$ , where the remainder is zero or has degree less than the divisor. This is the polynomial version of dividing integers. The quotient is not generally the whole answer: retain a non-zero remainder.

Arrange both expressions in descending powers. Insert zero coefficients for missing powers so that columns stay aligned. Divide the leading terms, multiply the divisor by that term, subtract the whole product, and repeat on the new leading term.

Choosing a method and checking it

Synthetic division is a compact coefficient method for a divisor $x - a$ . Use the root $a$ in the synthetic row; for $x + 2$ , the row uses $- 2$ . If the divisor is not monic, ordinary long division avoids a scaling error. A quadratic divisor may leave a linear remainder.

Check by multiplying the divisor by your quotient and adding the remainder. The result must reproduce every term of the original dividend. If you write a rational expression as $Q + R / D$ , remember it is defined only where the original denominator is non-zero.

A common trap

The remainder must always be a constant.

Instead: Its degree must be smaller than the divisor degree. A quadratic divisor can leave a linear remainder.

02 / WORKED EXAMPLES

See the thinking

Example 1

A linear divisor with a remainder

Divide $2 x^{3} + 3 x^{2} - 5 x + 6$ by $x + 2$ .

The first quotient term is $2 x^{2}$ . Subtract $(x + 2) 2 x^{2}$ to leave $- x^{2} - 5 x + 6$ .
The next term is $- x$ . Subtract $(x + 2) (- x)$ to leave $- 3 x + 6$ .
The last term is $- 3$ , leaving remainder $12$ . Check $(x + 2) (2 x^{2} - x - 3) + 12$ .

Quotient

2 x^{2} - x - 3

; remainder

12

Example 2

A quadratic divisor

Divide $x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 5$ by $x^{2} + 1$ .

Use leading quotient terms $x^{2}$ , then $2 x$ , then $2$ .
The product $(x^{2} + 1) (x^{2} + 2 x + 2)$ is $x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 2$ .
Subtract this product from the dividend to get $2 x + 3$ , whose degree is less than $2$ .

Quotient

x^{2} + 2 x + 2

; remainder

2 x + 3

Check your understanding

A degree- $5$ polynomial is divided by a degree- $2$ polynomial. What are the quotient degree and largest possible remainder degree?

Show answer

Quotient degree $3$ ; remainder degree at most $1$ .

Leading-term degrees subtract, while the remainder degree is strictly smaller than the divisor degree.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Divide $x^{3} - 6 x^{2} + 11 x - 6$ by $x - 1$ .
2 marks
Divide $2 x^{3} - 5 x^{2} + 4 x - 3$ by $x - 2$ .
2 marks
Divide $x^{4} - 16$ by $x^{2} + 4$ .
2 marks
Divide $x^{3} - 2 x + 1$ by $x + 1$ .
2 marks
Divide $3 x^{2} + x - 2$ by $3 x - 2$ .
2 marks
A student writes $x^{3} + 1 = (x + 1) (x^{2} - x) + 1$ . Check the claim and correct the division.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Divide $x^{3} - 6 x^{2} + 11 x - 6$ by $x - 1$ .
2 marks
Synthetic coefficients at $1$ are $1, - 5, 6, 0$ .
$Q = x^{2} - 5 x + 6$ , $R = 0$ .
Divide $2 x^{3} - 5 x^{2} + 4 x - 3$ by $x - 2$ .
2 marks
Synthetic coefficients at $2$ are $2, - 1, 2, 1$ .
$Q = 2 x^{2} - x + 2$ , $R = 1$ .
Divide $x^{4} - 16$ by $x^{2} + 4$ .
2 marks
Use the difference of squares: $(x^{2} - 4) (x^{2} + 4) = x^{4} - 16$ .
$Q = x^{2} - 4$ , $R = 0$ .
Divide $x^{3} - 2 x + 1$ by $x + 1$ .
2 marks
Include the missing $x^{2}$ coefficient: divide coefficients $1, 0, - 2, 1$ using $- 1$ .
$Q = x^{2} - x - 1$ , $R = 2$ .
Divide $3 x^{2} + x - 2$ by $3 x - 2$ .
2 marks
The leading term is $x$ ; the remaining term after subtraction is $3 x - 2$ .
$Q = x + 1$ , $R = 0$ .
A student writes $x^{3} + 1 = (x + 1) (x^{2} - x) + 1$ . Check the claim and correct the division.
3 marks
The claimed product plus remainder is $x^{3} - x + 1$ , not $x^{3} + 1$ . Adding $1$ to the quotient repairs the identity.
The correct quotient is $x^{2} - x + 1$ , remainder $0$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

A recognisable real-polynomial chapter supporting Stage 2 Specialist polynomial factorisation and graph analysis; original lessons use current SACE scope and the school chapter structure requested by the user.

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