What's a Polynomial? Degree and Family Tree — Class 9 Mathematics Chapter 2

🖼 Image PendingA vast cinematic 'family tree' rising into a starry sky — its trunk labelled 'Polynomial', branching into four glowing leaves: Constant, Linear, Quadratic, Cubic, each in a distinct colour

AI Generation Prompt

Ultra-wide cinematic banner (16:5 ratio). A glowing genealogical 'family tree' rises out of a dark misty foreground into a deep starry navy sky. The trunk is labelled with a soft golden 'POLYNOMIAL' inscription that fades into branches. Four broad branches arc outward, each ending in a stylised glowing leaf in a distinct colour: pale grey for 'Constant', emerald green for 'Linear', sky blue for 'Quadratic', violet for 'Cubic'. Faint mathematical formulas float around each leaf. The image conveys: every algebraic expression in one variable belongs to one of these branches by its degree. Painterly cinematic illustration in the style of mathematical wonder. Dark background. No text labels other than what is described.

✦Think About It

Look at four expressions: $4x$, $x^2 + 1$, $5y^3 + y^2 - 8$, $-9$. They all involve **one** variable. **Which is the simplest? Which is the most complicated? What rule could you use to compare their 'difficulty'?**

Notice the *highest power* of the variable in each — does that look like a useful ranking?

✦ब्रह्मस्फुटसिद्धान्त — ब्रह्मगुप्त (६२८ CE)

From Brahmagupta's Brahmasphuṭa-siddhānta — On Classifying Equations

एकवर्णसमीकरणम् वर्गसमीकरणं तथा

घनसमीकरणं चेति त्रिविधं समीक्ष्यताम्॥

(eka-varṇa-samīkaraṇam varga-samīkaraṇaṃ tathā / ghana-samīkaraṇaṃ ceti tri-vidhaṃ samīkṣyatām)

'समीकरण तीन प्रकार के होते हैं — सामान्य (एक चिन्ह), वर्ग (वर्गयुक्त), और घन (घनयुक्त)।'

'Equations are of three kinds — eka-varṇa (linear, one symbol), varga (quadratic, with a square), and ghana (cubic, with a cube).'

In 628 CE, Brahmagupta organised algebra by the highest power of the unknown — exactly the degree-based classification you'll learn on this page. The Sanskrit names — eka-varṇa for linear, varga for quadratic, ghana for cubic — appear in every Indian algebra text from then on. The English words quadratic (Latin quadratus, square) and cubic are direct translations of the Sanskrit. Brahmagupta named the family tree we're about to draw.

From algebraic expression to polynomial

On the previous page, you saw three kinds of expression — some with two variables ( $4x + 5y + 3$ , $200l + 160w + 50lw$ ) and one with just one ( $10x - x^2$ ). From now on, we focus on the one-variable case.

A polynomial in one variable (also called a univariate polynomial, or just polynomial when the context is clear) is an expression built from a single variable, its non-negative integer powers, and numbers — combined with addition, subtraction, and multiplication.

Examples that are polynomials:

$4x$ · $x^2 + 5x + 1$ · $2y - 5$ · $5y^3 + y^2 + 2y - 1$ · $3z + 7$ · $-9$

Examples that are not polynomials:

$\sqrt{x} + 1$ — it has $x^{1/2}$ , a fractional power. Not allowed.
$\frac{1}{x} + 5$ — it has $x^{-1}$ , a negative power. Not allowed.
$2^x$ — the variable is in the exponent, not the base. Not allowed.

In a polynomial, the variable's only powers are 0, 1, 2, 3, 4, … (whole numbers ≥ 0).

Degree — the highest power that appears

Every polynomial has a degree — the highest power of the variable that appears in any term. The degree is the single most important number describing a polynomial.

In $x^2 + 5x + 1$ , the highest power of $x$ is 2. Degree = 2.
In $5y^3 + y^2 - 8$ , the highest power of $y$ is 3. Degree = 3.
In $3z + 7 = 3z^1 + 7z^0$ , the highest power of $z$ is 1. Degree = 1.
In $8$ , there is no variable — but we can write $8 = 8x^0$ , so the highest power is 0. Degree = 0.

Degree organises polynomials into families:

Degree	Family name	Example
0	constant polynomial	$8$ , $-9$ , $\frac{1}{2}$
1	linear polynomial	$3z + 7$ , $4x$ , $2n - 1$
2	quadratic polynomial	$x^2 + 5x + 1$ , $7s^2 - 4s + 6$
3	cubic polynomial	$5y^3 + y^2 + 2y - 1$

Degrees 4, 5, 6 and higher exist too — they don't have such friendly names, but you'll meet them later. The rest of this chapter zooms entirely into the linear case (degree 1) — the simplest non-trivial family.

Loading simulator…

Worked Example 1Find the degree (NCERT Ex Set 2.1, Q1)

NCERT

Find the degrees of: (i) $2x^2 - 5x + 3$ (ii) $y^3 + 2y - 1$ (iii) $-9$ (iv) $4z - 3$

(i) $2x^2 - 5x + 3$ — highest power of $x$ is 2. Degree = 2 (quadratic).

(ii) $y^3 + 2y - 1$ — highest power of $y$ is 3. Degree = 3 (cubic).

(iii) $-9$ — no variable; write as $-9 \cdot x^0$ . Degree = 0 (constant).

(iv) $4z - 3$ — highest power of $z$ is 1. Degree = 1 (linear).

Worked Example 2Write polynomials of given degrees (NCERT Q2)

NCERT

Write polynomials of degrees 1, 2, and 3.

Many answers are possible. Some examples:

Degree 1 (linear): $3x + 7$ , or $-2y + 1$ , or $5z$ .
Degree 2 (quadratic): $x^2 + 4x + 4$ , or $-y^2 + 1$ , or $7s^2$ .
Degree 3 (cubic): $x^3 - x$ , or $2y^3 + y^2 + 5$ , or $z^3$ .

Rule: the highest power of the variable must be exactly the degree you want — no higher. So for a linear polynomial, you cannot include any $x^2$ or $x^3$ term.

Worked Example 3Coefficients (NCERT Q3)

NCERT

What are the coefficients of $x^2$ and $x^3$ in the polynomial $x^4 - 3x^3 + 6x^2 - 2x + 7$ ?

Step 1 — Pick out the term containing $x^2$ . It is $6x^2$ . Coefficient of $x^2$ is $6$ .

Step 2 — Pick out the term containing $x^3$ . It is $-3x^3$ . Coefficient of $x^3$ is $-3$ . (The negative sign belongs with the coefficient.)

Bonus observations: the polynomial has degree 4 (highest power $x^4$ , coefficient $1$ ); the constant term is $7$ ; the coefficient of $x$ is $-2$ .

Worked Example 4Coefficient of z (NCERT Q4)

NCERT

What is the coefficient of $z$ in the polynomial $4z^3 + 5z^2 - 11$ ?

Look for the term containing $z^1$ . There isn't one. The polynomial is $4z^3 + 5z^2 + 0 \cdot z - 11$ . So the coefficient of $z$ is $0$ .

Lesson: a coefficient of zero means the term is missing. Coefficients exist for every power up to the degree, but some can be 0.

Worked Example 5Constant term (NCERT Q5)

NCERT

What is the constant term of the polynomial $9x^3 + 5x^2 - 8x - 10$ ?

The constant term is the one without any variable. Here it is $-10$ . (Sign included.)

Quick observation: The constant term is also the value of the polynomial at $x = 0$ . Substitute $x = 0$ : $9(0) + 5(0) - 8(0) - 10 = -10$ . ✓ Always true: constant term = value at $x = 0$ .

Logical ReasoningLevel 3 · Analysis

Is $\sqrt{x} + 1$ a polynomial? Is $\frac{1}{x} + 5$? **Why or why not?**

Practice Yourself — Identify, Classify, Inspect

For each polynomial: state the degree, the family name, and the coefficient of the highest-power term (the leading coefficient). Cover the answers and try.

$7x + 4$
$x^2 - 9$
$-3y^3 + y$
$5$
$2x^4 - x^3 + x - 1$
$-z^2$
$\frac{1}{2}x + 3$
$11y$
$x^3 + x^2 + x + 1$
$-7$

Answers: 1. degree 1, linear, leading coeff 7. 2. degree 2, quadratic, leading coeff 1. 3. degree 3, cubic, leading coeff −3. 4. degree 0, constant, the constant is the leading coeff = 5. 5. degree 4, no friendly name (call it 'degree-4 polynomial'), leading coeff 2. 6. degree 2, quadratic, leading coeff −1. 7. degree 1, linear, leading coeff ½. 8. degree 1, linear, leading coeff 11. 9. degree 3, cubic, leading coeff 1. 10. degree 0, constant, leading coeff −7.

Quick Check

Q1.What is the degree of the polynomial $4z - 3$ ?

AI Generation Prompt

✦Think About It

Notice the *highest power* of the variable in each — does that look like a useful ranking?

✦ब्रह्मस्फुटसिद्धान्त — ब्रह्मगुप्त (६२८ CE)

From Brahmagupta's Brahmasphuṭa-siddhānta — On Classifying Equations

एकवर्णसमीकरणम् वर्गसमीकरणं तथा

घनसमीकरणं चेति त्रिविधं समीक्ष्यताम्॥

(eka-varṇa-samīkaraṇam varga-samīkaraṇaṃ tathā / ghana-samīkaraṇaṃ ceti tri-vidhaṃ samīkṣyatām)

'Equations are of three kinds — eka-varṇa (linear, one symbol), varga (quadratic, with a square), and ghana (cubic, with a cube).'

From algebraic expression to polynomial

Examples that are polynomials:

$4x$ · $x^2 + 5x + 1$ · $2y - 5$ · $5y^3 + y^2 + 2y - 1$ · $3z + 7$ · $-9$

Examples that are not polynomials:

$\sqrt{x} + 1$ — it has $x^{1/2}$ , a fractional power. Not allowed.
$\frac{1}{x} + 5$ — it has $x^{-1}$ , a negative power. Not allowed.
$2^x$ — the variable is in the exponent, not the base. Not allowed.

In a polynomial, the variable's only powers are 0, 1, 2, 3, 4, … (whole numbers ≥ 0).

Degree — the highest power that appears

Every polynomial has a degree — the highest power of the variable that appears in any term. The degree is the single most important number describing a polynomial.

In $x^2 + 5x + 1$ , the highest power of $x$ is 2. Degree = 2.
In $5y^3 + y^2 - 8$ , the highest power of $y$ is 3. Degree = 3.
In $3z + 7 = 3z^1 + 7z^0$ , the highest power of $z$ is 1. Degree = 1.
In $8$ , there is no variable — but we can write $8 = 8x^0$ , so the highest power is 0. Degree = 0.

Degree organises polynomials into families:

Degree	Family name	Example
0	constant polynomial	$8$ , $-9$ , $\frac{1}{2}$
1	linear polynomial	$3z + 7$ , $4x$ , $2n - 1$
2	quadratic polynomial	$x^2 + 5x + 1$ , $7s^2 - 4s + 6$
3	cubic polynomial	$5y^3 + y^2 + 2y - 1$

Loading simulator…

Worked Example 1Find the degree (NCERT Ex Set 2.1, Q1)

NCERT

Find the degrees of: (i) $2x^2 - 5x + 3$ (ii) $y^3 + 2y - 1$ (iii) $-9$ (iv) $4z - 3$

(i) $2x^2 - 5x + 3$ — highest power of $x$ is 2. Degree = 2 (quadratic).

(ii) $y^3 + 2y - 1$ — highest power of $y$ is 3. Degree = 3 (cubic).

(iii) $-9$ — no variable; write as $-9 \cdot x^0$ . Degree = 0 (constant).

(iv) $4z - 3$ — highest power of $z$ is 1. Degree = 1 (linear).

Worked Example 2Write polynomials of given degrees (NCERT Q2)

NCERT

Write polynomials of degrees 1, 2, and 3.

Many answers are possible. Some examples:

Degree 1 (linear): $3x + 7$ , or $-2y + 1$ , or $5z$ .
Degree 2 (quadratic): $x^2 + 4x + 4$ , or $-y^2 + 1$ , or $7s^2$ .
Degree 3 (cubic): $x^3 - x$ , or $2y^3 + y^2 + 5$ , or $z^3$ .

Rule: the highest power of the variable must be exactly the degree you want — no higher. So for a linear polynomial, you cannot include any $x^2$ or $x^3$ term.

Worked Example 3Coefficients (NCERT Q3)

NCERT

What are the coefficients of $x^2$ and $x^3$ in the polynomial $x^4 - 3x^3 + 6x^2 - 2x + 7$ ?

Step 1 — Pick out the term containing $x^2$ . It is $6x^2$ . Coefficient of $x^2$ is $6$ .

Step 2 — Pick out the term containing $x^3$ . It is $-3x^3$ . Coefficient of $x^3$ is $-3$ . (The negative sign belongs with the coefficient.)

Bonus observations: the polynomial has degree 4 (highest power $x^4$ , coefficient $1$ ); the constant term is $7$ ; the coefficient of $x$ is $-2$ .

Worked Example 4Coefficient of z (NCERT Q4)

NCERT

What is the coefficient of $z$ in the polynomial $4z^3 + 5z^2 - 11$ ?

Look for the term containing $z^1$ . There isn't one. The polynomial is $4z^3 + 5z^2 + 0 \cdot z - 11$ . So the coefficient of $z$ is $0$ .

Lesson: a coefficient of zero means the term is missing. Coefficients exist for every power up to the degree, but some can be 0.

Worked Example 5Constant term (NCERT Q5)

NCERT

What is the constant term of the polynomial $9x^3 + 5x^2 - 8x - 10$ ?

The constant term is the one without any variable. Here it is $-10$ . (Sign included.)

Quick observation: The constant term is also the value of the polynomial at $x = 0$ . Substitute $x = 0$ : $9(0) + 5(0) - 8(0) - 10 = -10$ . ✓ Always true: constant term = value at $x = 0$ .

Logical ReasoningLevel 3 · Analysis

Is $\sqrt{x} + 1$ a polynomial? Is $\frac{1}{x} + 5$? **Why or why not?**

Quick Check

Q1.What is the degree of the polynomial $4z - 3$ ?