Polynomial notation and operations | Year 12

01 / LEARN

Understand the idea.

What makes an expression a polynomial

A real polynomial is a finite sum of real coefficients times nonnegative integer powers of $x$ . The coefficients may be negative, fractional or irrational; it is the powers of the variable that are restricted.

Write terms in descending powers. The largest exponent with a nonzero coefficient is the degree, and its coefficient is the leading coefficient. Missing powers have coefficient zero. A nonzero constant has degree zero; the zero polynomial has no ordinary degree.

Operate by powers, then check

Collect like powers when adding or subtracting. A minus sign before a bracket changes every sign inside it. When multiplying, distribute each term across the other polynomial before combining terms of the same power.

For nonzero factors, product degrees add because their leading coefficients multiply to a nonzero value. In a sum, leading terms can cancel. Check your expansion by degree and substitution; a single matching value is a check, not an identity proof.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Subtracting a whole polynomial

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

Worked solution

1
Distribute the subtraction to every term.
$\begin{aligned} (3 x^{3} - 2 x + 5) & - (x^{3} + 4 x^{2} - x - 7) \\ = 3 x^{3} - 2 x + 5 - x^{3} - 4 x^{2} + x + 7. \end{aligned}$
Subtracting a polynomial changes the sign of every term in its bracket.
2
Group terms with matching powers.
$(3 - 1) x^{3} - 4 x^{2} + (- 2 + 1) x + (5 + 7) .$
Only terms with the same power of $x$ have coefficients that can be combined.
3
Simplify the coefficients.
$2 x^{3} - 4 x^{2} - x + 12.$
The cubic, quadratic, linear and constant coefficients are respectively $2, - 4, - 1, 12$ .

Answer

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

EXAMPLE 02

Multiplying two factors

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

Worked solution

1
Multiply the quadratic by each term.
$\begin{aligned} (2 x - 3) (x^{2} + x - 4) & = 2 x (x^{2} + x - 4) \\ - 3 (x^{2} + x - 4) . \end{aligned}$
The distributive law applies to both terms of the linear factor.
2
Expand both contributions.
$2 x^{3} + 2 x^{2} - 8 x - 3 x^{2} - 3 x + 12.$
Multiplying powers adds their exponents, and each coefficient keeps its sign.
3
Collect like powers.
$2 x^{3} - x^{2} - 11 x + 12.$
The result has degree $1 + 2 = 3$ , consistent with the nonzero factors.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Yes; degree $5$ .

The exponents in $4 - \frac{1}{2} x^{5}$ are zero and five, both nonnegative integers. A fractional coefficient is allowed, so it is a polynomial of degree five.

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 $3$ .

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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.

Official curriculum source ↗

In this lesson

Understand the idea.

What makes an expression a polynomial

Operate by powers, then check

See the method in action.

Subtracting a whole polynomial

Distribute the subtraction to every term.

Group terms with matching powers.

Simplify the coefficients.

Multiplying two factors

Multiply the quadratic by each term.

Expand both contributions.

Collect like powers.

Try it yourself.

Check your understanding

Keep practising