Polynomial notation and operations | Year 12

01 / LEARN

The idea, step by step

What makes an expression a polynomial

A real polynomial in $x$ is a finite sum of terms $a_{n} x^{n}$ in which every exponent is a non-negative integer and every coefficient is real. Write its terms in descending powers to expose the degree and leading coefficient. Missing powers have coefficient zero; a non-zero constant has degree zero.

The restriction is on the variable, not on the coefficient: $\sqrt{2} x^{3} - \frac{1}{2} x + 4$ is a polynomial in $x$ . Expressions such as $x^{- 1}$ , $\sqrt{x}$ and $\frac{1}{x + 2}$ are not polynomials in $x$ . The zero polynomial has no ordinary degree.

Operate by powers, then check

Addition and subtraction combine coefficients of matching powers. Put a subtracted polynomial in brackets first so that the minus sign changes every term. For multiplication, distribute each term across the other factor; only then collect like powers.

If non-zero polynomials of degrees $m$ and $n$ are multiplied, the product has degree $m + n$ . A sum can have lower degree when leading terms cancel. Degree and a quick substitution check can reveal an omitted term, although one numerical check alone cannot prove two polynomials identical.

A common trap

The degree is the number of terms, or every power must appear.

Instead: The degree is the largest exponent with non-zero coefficient. Missing powers are allowed.

02 / WORKED EXAMPLES

See the thinking

Example 1

Subtracting a whole polynomial

Simplify $(3 x^{3} - 2 x + 5) - (x^{3} + 4 x^{2} - x - 7)$ .

Remove the second bracket carefully: $3 x^{3} - 2 x + 5 - x^{3} - 4 x^{2} + x + 7$ .
Collect equal powers: $(3 - 1) x^{3} - 4 x^{2} + (- 2 + 1) x + (5 + 7)$ .

2 x^{3} - 4 x^{2} - x + 12

Example 2

Multiplying two factors

Expand $(2 x - 3) (x^{2} + x - 4)$ .

Distribute: $2 x^{3} + 2 x^{2} - 8 x - 3 x^{2} - 3 x + 12$ .
Collect terms: $2 x^{3} - x^{2} - 11 x + 12$ . The leading term has degree $1 + 2 = 3$ .

2 x^{3} - x^{2} - 11 x + 12

Check your understanding

Is $4 - \frac{1}{2} x^{5}$ a polynomial? State its degree.

Show answer

Yes; degree $5$ .

The exponent is a non-negative integer; a fractional coefficient is allowed.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

State the degree and leading coefficient of $7 - 2 x^{4} + x$ .
2 marks
Which are polynomials in $x$ : $x^{2} + \sqrt{3}$ , $x^{- 2} + 1$ , $\sqrt{x} + 2$ ?
2 marks
Add $4 x^{3} - x^{2} + 6$ and $- x^{3} + 3 x - 8$ .
2 marks
Simplify $(2 x^{2} - 5 x + 1) - (3 x^{2} + x - 4)$ .
2 marks
Expand $(x + 2) (x^{2} - 3 x + 5)$ .
2 marks
A non-zero cubic is multiplied by a non-zero quartic. What is the degree of the product? Can their sum have degree $3$ ?
2 marks

Read the worked solutions

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

State the degree and leading coefficient of $7 - 2 x^{4} + x$ .
2 marks
In descending order it is $- 2 x^{4} + x + 7$ .
Degree $4$ ; leading coefficient $- 2$ .
Which are polynomials in $x$ : $x^{2} + \sqrt{3}$ , $x^{- 2} + 1$ , $\sqrt{x} + 2$ ?
2 marks
A constant $\sqrt{3}$ is allowed. Negative or fractional powers of $x$ are not.
Only $x^{2} + \sqrt{3}$ .
Add $4 x^{3} - x^{2} + 6$ and $- x^{3} + 3 x - 8$ .
2 marks
Add coefficients of each matching power, including zero coefficients.
$3 x^{3} - x^{2} + 3 x - 2$
Simplify $(2 x^{2} - 5 x + 1) - (3 x^{2} + x - 4)$ .
2 marks
Change all signs in the second bracket, then collect like terms.
$- x^{2} - 6 x + 5$
Expand $(x + 2) (x^{2} - 3 x + 5)$ .
2 marks
$x^{3} - 3 x^{2} + 5 x + 2 x^{2} - 6 x + 10 = x^{3} - x^{2} - x + 10$ .
$x^{3} - x^{2} - x + 10$
A non-zero cubic is multiplied by a non-zero quartic. What is the degree of the product? Can their sum have degree $3$ ?
2 marks
Multiply leading terms to obtain degree $3 + 4$ . The cubic has no fourth-degree term to cancel the quartic leading term.
Product degree $7$ ; their sum has degree $4$ , not $3$ .

Checking saved progress…

Continue learning

Zeros, roots and factors

Next lesson

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.

Official curriculum source ↗

Back to Real Polynomials