SPI Mathematics · Polynomials

Degree, Value and Zeroes of a Polynomial

The degree of a polynomial is its highest power of x, its value at a number comes from putting that number in for x, and a zero is a value of x that makes it 0.

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Why this matters

Eight questions in the bank, seven of them EASY: name the degree, pick the linear polynomial, find a value, or count the zeroes on a graph. These are the quickest marks in the chapter, and they are lost to a squared negative number or to counting where a graph meets the wrong axis.

Concept 1 of 3: Degree of a polynomial and the name for each degree

A polynomial is a sum of terms like

5x^3

-2x

or 7, where every power of x is a whole number. The highest power present is its degree, and each low degree has its own name.

Definition

The degree is the highest power of x with a non-zero coefficient, wherever that term is written.
Every power must be a whole number. $x + \frac{1}{x}$ is not a polynomial (it has $x^{-1}$ ), and neither is anything with $\sqrt{x}$ .
A number such as 7 or $\sqrt{2}$ is a constant term: its degree is 0, so it never changes the degree.

Name	Degree	Example
Constant polynomial	0	7
Linear polynomial	1	3x + 2
Quadratic polynomial	2	x² − 4x + 1
Cubic polynomial	3	2x³ + x − 5
Biquadratic (quartic) polynomial	4	x⁴ − 5x² + 6 A biquadratic is a quadratic in x², so its degree is 4, not 2.

The same names are used for equations: a cubic equation has degree 3.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Polynomials · Degree, Value and Zeroes of a Polynomial

Which of the following is a linear polynomial?

Taking the degree from the first term

The degree is the highest power anywhere in the polynomial.

2x - x^3 + 4

has degree 3, even though it starts with an x term.

Treating a root sign as a power of x

\sqrt{2}

is a constant and does not change the degree.

\sqrt{x}

x^{1/2}

, which makes the expression not a polynomial.

Drill 3 more on degree of a polynomial and the name for each degree

Concept 2 of 3: Value of a polynomial and what a zero is

To find the value of a polynomial at a number, put that number in place of x and work it out. If the answer is 0, that number is a zero of the polynomial.

Definition

$p(a)$ means the value of $p(x)$ when $x = a$ .
a is a zero of $p(x)$ when $p(a) = 0$ .
Put a negative number in brackets and square it before multiplying by the coefficient: $-3(-2)^2 = -3 \times 4 = -12$ .
The zero of a linear polynomial $ax + b$ is $-\frac{b}{a}$ .

Zero of a polynomial

p(a) = 0 \qquad \text{zero of } ax + b = -\frac{b}{a}

Worked example

For

p(x) = 2x^2 - 3x + 1

, find

p(-2)

and check whether 1 is a zero.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Polynomials · Degree, Value and Zeroes of a Polynomial

The value of the polynomial

5x - 4x^2 + 3

, when

x = -1

Squaring after attaching the minus sign

x = -2

, the term

-3x^2

-3 \times 4 = -12

, not

+12

. Square the number first, then multiply by the coefficient with its sign.

Drill 1 more on value of a polynomial and what a zero is

Concept 3 of 3: Reading the zeroes of a polynomial from its graph

The graph of

y = p(x)

meets the x-axis exactly where

p(x) = 0

. So the zeroes are the x-values of the points where the curve cuts or touches the x-axis, and the number of zeroes is the number of such points.

Definition

A polynomial of degree n has at most n zeroes.
The graph of a quadratic polynomial is a parabola, one U shape: it opens upwards when the coefficient of $x^2$ is positive and downwards when it is negative.
A curve that rises, falls and rises again, crossing the x-axis three times, is a cubic, not a quadratic.
Points on the y-axis are not zeroes. Only the x-axis counts.

What the graph does	Number of distinct zeroes
A straight line crosses the x-axis once	1
A parabola cuts the x-axis at two points	2
A parabola only touches the x-axis at one point	1 Touching counts: the point where it touches is a zero.
A parabola never meets the x-axis	0
A cubic curve crosses the x-axis three times	3

Count the meeting points with the x-axis, not with the y-axis.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Polynomials · Degree, Value and Zeroes of a Polynomial

In the given figure, graph of

p(x)

is shown. Number of distinct zeroes of

p(x)

is:

Counting where the graph meets the y-axis

Zeroes are where the graph meets the x-axis. A curve that cuts the y-axis at 3 has

p(0) = 3

, which says nothing about its zeroes.

Missing a zero where the curve only touches

A parabola that just touches the x-axis and turns back still meets it there, so it has exactly one distinct zero, not none.

Drill 1 more on reading the zeroes of a polynomial from its graph

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Value of a polynomial and what a zero is
Zero of a polynomial
$p(a) = 0 \qquad \text{zero of } ax + b = -\frac{b}{a}$

Reference tables (2)

Degree of a polynomial and the name for each degree5 rows

Name	Degree	Example
Constant polynomial	0	7
Linear polynomial	1	3x + 2
Quadratic polynomial	2	x² − 4x + 1
Cubic polynomial	3	2x³ + x − 5
Biquadratic (quartic) polynomial	4	x⁴ − 5x² + 6 A biquadratic is a quadratic in x², so its degree is 4, not 2.

The same names are used for equations: a cubic equation has degree 3.

Reading the zeroes of a polynomial from its graph5 rows

What the graph does	Number of distinct zeroes
A straight line crosses the x-axis once	1
A parabola cuts the x-axis at two points	2
A parabola only touches the x-axis at one point	1 Touching counts: the point where it touches is a zero.
A parabola never meets the x-axis	0
A cubic curve crosses the x-axis three times	3

Count the meeting points with the x-axis, not with the y-axis.

Watch out for (5)

Taking the degree from the first term
→ Degree of a polynomial and the name for each degree
Treating a root sign as a power of x
→ Degree of a polynomial and the name for each degree
Squaring after attaching the minus sign
→ Value of a polynomial and what a zero is
Counting where the graph meets the y-axis
→ Reading the zeroes of a polynomial from its graph
Missing a zero where the curve only touches
→ Reading the zeroes of a polynomial from its graph

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