Linear Polynomials in the Wild — Class 9 Mathematics Chapter 2

🖼 Image PendingA wide cinematic view of an evenly-spaced ladder of glowing rungs ascending into a deep starry sky, each rung labelled with a number — the rungs march upward in equal steps, the visual signature of a linear pattern

AI Generation Prompt

Ultra-wide cinematic banner (16:5 ratio). A vertical glowing ladder rising into a deep starry navy sky. Each rung is evenly spaced from the next — the ladder ascends in clean, equal steps. The rungs themselves are luminous bars of warm amber-gold light. Faint numerical labels float beside each rung — '250, 300, 350, 400, …' — capturing the constant-difference signature of a linear pattern. The image conveys: every linear polynomial is a ladder where each step is the same size. Painterly cinematic illustration. Dark background. No additional text labels.

✦Think About It

Look at this list of numbers: **250, 300, 350, 400, 450, …** What's the next number? What's the **100th** number? **And how are you so sure?**

What's the difference between consecutive numbers in the list? Is it the same every time?

✦सूत्रात्मक सिद्धांत — आर्यभटीय (४९९ CE)

From Āryabhaṭa's tradition — On Steady Increase

एकैकं वर्धमानानां योगः सर्वत्र सिद्ध्यति

सम-वृद्ध्या यथा कालः, तथा सङ्ख्या सम-गत्या॥

(ekaikaṃ vardhamānānāṃ yogaḥ sarvatra siddhyati / sama-vṛddhyā yathā kālaḥ, tathā saṃkhyā sama-gatyā)

'जो चीज़ बराबर-बराबर बढ़ती है, उसका जोड़ हमेशा निकाला जा सकता है। जैसे समय बराबर बहता है, वैसे ही ऐसी संख्याएँ बराबर चलती हैं।'

'For things that increase by equal amounts, the sum can always be found. Just as time flows in equal beats, such numbers march in equal steps.'

Āryabhaṭa (499 CE) and his successors knew that equal-step sequences (what we now call arithmetic progressions or linear patterns) had simple closed-form expressions. The same idea — constant difference between consecutive terms — became the signature of linearity in algebra. You'll meet linear patterns again as 'arithmetic progressions' in a later chapter; today, you're meeting their algebraic form.

A linear polynomial — degree 1, the friendliest family

On the previous page you met four polynomial families. Of these, the linear family — degree 1 — is by far the most common in everyday life. A linear polynomial in $x$ is anything of the form

$ax + b$

where $a$ and $b$ are numbers and $a \neq 0$ (otherwise it would just be the constant $b$ , not a linear polynomial).

$a$ is the leading coefficient (the number multiplying $x$ ).
$b$ is the constant term (the number standing alone).
The variable can be any letter — $x$ , $y$ , $z$ , $m$ , $n$ , $t$ , … the name doesn't change anything.

Familiar linear polynomials:

$4x$ (perimeter of a square of side $x$ ) — here $a = 4$ , $b = 0$ .
$2x + 3$ (the function-machine example) — $a = 2$ , $b = 3$ .
$200 + 50m$ (chess-club fee for $m$ matches) — $a = 50$ , $b = 200$ .
$4z - 3$ — $a = 4$ , $b = -3$ .

Worked Example 1Square perimeter (NCERT Ex 4)

NCERT

The perimeter of a square of side $x$ is $4x$ . Fill in the perimeters for sides $1, 1.5, 2, 2.5, 3$ cm. What happens to the perimeters as the side increases by 0.5 cm each time?

Step 1 — Substitute each side length into $P = 4x$ .

Side $x$ (cm)	1	1.5	2	2.5	3
Perimeter $4x$ (cm)	4	6	8	10	12

Step 2 — Spot the pattern.

Each time $x$ goes up by 0.5, the perimeter goes up by $4 \times 0.5 = 2$ cm. Looking at consecutive perimeters: $6 - 4 = 2$ , $8 - 6 = 2$ , $10 - 8 = 2$ , $12 - 10 = 2$ . The successive differences are constant — exactly equal to the leading coefficient, $4$ , times the step size.

Answer: Perimeters increase by exactly 2 cm for every 0.5 cm the side grows.

Worked Example 2Chess club fee (NCERT Ex 5)

NCERT

A chess club charges ₹200 to join, plus ₹50 for every match played. Build a table for 1, 2, 3, 4, 5, …, $m$ matches. Express the cost as a linear polynomial in $m$ .

Step 1 — Build the table.

Matches	1	2	3	4	5	…	$m$
Amount paid (₹)	250	300	350	400	450	…	$200 + 50m$

Step 2 — Spot the pattern.

For 1 match: ₹200 (joining) + ₹50 (1 match) = ₹250. For 2 matches: ₹200 + ₹100 = ₹300. Etc. For $m$ matches: $200 + 50m$ .

Step 3 — Identify the linear polynomial.

$\text{Total} = 200 + 50m = 50m + 200$

This is linear ( $a = 50$ , $b = 200$ ). Successive differences are exactly $50$ (one extra match = ₹50 extra), the leading coefficient. ✓

Answer: Cost = $50m + 200$ rupees.

Worked Example 3How many matches did he play? (NCERT Think-Reflect)

NCERT

If a chess-club player paid ₹750, how many matches did he play?

Set the linear polynomial equal to ₹750:

$50m + 200 = 750$

Subtract 200:

$50m = 550$

Divide by 50:

$m = 11$

Answer: he played 11 matches.

(This is your first linear equation in this chapter — Page 5 will set up the formal recipe for solving them.)

Worked Example 4Coefficient of z (NCERT Ex Set 2.1, Q4)

NCERT

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

Look for the term containing $z^1 = z$ . There isn't one. So the coefficient of $z$ is $0$ .

Writing out all coefficients of $4z^3 + 5z^2 - 11$ :

Coefficient of $z^3$ : $4$
Coefficient of $z^2$ : $5$
Coefficient of $z$ : $0$
Constant: $-11$

When a power of the variable is missing from a polynomial, its coefficient is implicitly zero.

Worked Example 5Constant term (NCERT Q5)

NCERT

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

The constant is the term without any variable. Here it is $-10$ .

Cross-check: the constant term is always equal to the value of the polynomial at $x = 0$ . Substituting:

$9(0)^3 + 5(0)^2 - 8(0) - 10 = -10. ✓$

This is a handy mental trick — to find the constant term of any polynomial, just plug in $x = 0$ .

Loading simulator…

The constant-difference signature of every linear pattern

Both Examples 4 (square perimeter) and 5 (chess fee) shared one striking feature: the values jumped by a constant amount each time the variable went up by 1. This is no coincidence. It's the defining property of a linear polynomial.

The rule. For any linear polynomial $f(n) = an + b$ :

$f(n+1) - f(n) = [a(n+1) + b] - [an + b] = a$

The successive differences are always equal to the leading coefficient $a$ . Always. No matter where you are on the line.

This gives a test for linearity:

Compute $f(1), f(2), f(3), f(4), \ldots$
Look at the differences $f(2) - f(1)$ , $f(3) - f(2)$ , etc.
If all differences are equal → the pattern is linear.
If the differences themselves change (like for $f(n) = n^2$ , where they go $3, 5, 7, 9, \ldots$ ) → not linear.

The simulator above lets you toggle a comparison with $n^2$ to see this difference visually.

Logical ReasoningLevel 3 · Analysis

Look at the sequence $1, 4, 9, 16, 25, 36, \ldots$. Each number is the square of its position: $1 = 1^2$, $4 = 2^2$, etc. **Is this a linear pattern? Apply the constant-difference test.**

Practice Yourself — Spot Linear Patterns

For each sequence, check whether it's a linear pattern. If it is, find the constant difference $a$ and the linear polynomial $f(n) = an + b$ that produces it (set $f(1)$ = first term).

$5, 8, 11, 14, 17, \ldots$
$100, 90, 80, 70, 60, \ldots$
$2, 4, 8, 16, 32, \ldots$
$0, 7, 14, 21, 28, \ldots$
$1, 1, 1, 1, 1, \ldots$
$3, 5, 9, 15, 23, \ldots$
$-2, 3, 8, 13, 18, \ldots$
$1000, 950, 900, 850, 800, \ldots$

Answers: 1. Linear, $a = 3$ , $f(n) = 3n + 2$ . 2. Linear (decay), $a = -10$ , $f(n) = -10n + 110$ . 3. Not linear — differences are $2, 4, 8, 16$ , growing. (It's geometric.) 4. Linear, $a = 7$ , $f(n) = 7n - 7$ . 5. Linear (constant), $a = 0$ , $f(n) = 0 \cdot n + 1 = 1$ . 6. Not linear — differences are $2, 4, 6, 8$ , growing. 7. Linear, $a = 5$ , $f(n) = 5n - 7$ . 8. Linear (decay), $a = -50$ , $f(n) = -50n + 1050$ .

Quick Check

Q1.What is the leading coefficient $a$ in the linear polynomial $f(m) = 50m + 200$ ?

AI Generation Prompt

✦Think About It

Look at this list of numbers: **250, 300, 350, 400, 450, …** What's the next number? What's the **100th** number? **And how are you so sure?**

What's the difference between consecutive numbers in the list? Is it the same every time?

✦सूत्रात्मक सिद्धांत — आर्यभटीय (४९९ CE)

From Āryabhaṭa's tradition — On Steady Increase

एकैकं वर्धमानानां योगः सर्वत्र सिद्ध्यति

सम-वृद्ध्या यथा कालः, तथा सङ्ख्या सम-गत्या॥

(ekaikaṃ vardhamānānāṃ yogaḥ sarvatra siddhyati / sama-vṛddhyā yathā kālaḥ, tathā saṃkhyā sama-gatyā)

'For things that increase by equal amounts, the sum can always be found. Just as time flows in equal beats, such numbers march in equal steps.'

A linear polynomial — degree 1, the friendliest family

On the previous page you met four polynomial families. Of these, the linear family — degree 1 — is by far the most common in everyday life. A linear polynomial in $x$ is anything of the form

$ax + b$

where $a$ and $b$ are numbers and $a \neq 0$ (otherwise it would just be the constant $b$ , not a linear polynomial).

$a$ is the leading coefficient (the number multiplying $x$ ).
$b$ is the constant term (the number standing alone).
The variable can be any letter — $x$ , $y$ , $z$ , $m$ , $n$ , $t$ , … the name doesn't change anything.

Familiar linear polynomials:

$4x$ (perimeter of a square of side $x$ ) — here $a = 4$ , $b = 0$ .
$2x + 3$ (the function-machine example) — $a = 2$ , $b = 3$ .
$200 + 50m$ (chess-club fee for $m$ matches) — $a = 50$ , $b = 200$ .
$4z - 3$ — $a = 4$ , $b = -3$ .

Worked Example 1Square perimeter (NCERT Ex 4)

NCERT

The perimeter of a square of side $x$ is $4x$ . Fill in the perimeters for sides $1, 1.5, 2, 2.5, 3$ cm. What happens to the perimeters as the side increases by 0.5 cm each time?

Step 1 — Substitute each side length into $P = 4x$ .

Side $x$ (cm)	1	1.5	2	2.5	3
Perimeter $4x$ (cm)	4	6	8	10	12

Step 2 — Spot the pattern.

Answer: Perimeters increase by exactly 2 cm for every 0.5 cm the side grows.

Worked Example 2Chess club fee (NCERT Ex 5)

NCERT

A chess club charges ₹200 to join, plus ₹50 for every match played. Build a table for 1, 2, 3, 4, 5, …, $m$ matches. Express the cost as a linear polynomial in $m$ .

Step 1 — Build the table.

Matches	1	2	3	4	5	…	$m$
Amount paid (₹)	250	300	350	400	450	…	$200 + 50m$

Step 2 — Spot the pattern.

For 1 match: ₹200 (joining) + ₹50 (1 match) = ₹250. For 2 matches: ₹200 + ₹100 = ₹300. Etc. For $m$ matches: $200 + 50m$ .

Step 3 — Identify the linear polynomial.

$\text{Total} = 200 + 50m = 50m + 200$

This is linear ( $a = 50$ , $b = 200$ ). Successive differences are exactly $50$ (one extra match = ₹50 extra), the leading coefficient. ✓

Answer: Cost = $50m + 200$ rupees.

Worked Example 3How many matches did he play? (NCERT Think-Reflect)

NCERT

If a chess-club player paid ₹750, how many matches did he play?

Set the linear polynomial equal to ₹750:

$50m + 200 = 750$

Subtract 200:

$50m = 550$

Divide by 50:

$m = 11$

Answer: he played 11 matches.

(This is your first linear equation in this chapter — Page 5 will set up the formal recipe for solving them.)

Worked Example 4Coefficient of z (NCERT Ex Set 2.1, Q4)

NCERT

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

Look for the term containing $z^1 = z$ . There isn't one. So the coefficient of $z$ is $0$ .

Writing out all coefficients of $4z^3 + 5z^2 - 11$ :

Coefficient of $z^3$ : $4$
Coefficient of $z^2$ : $5$
Coefficient of $z$ : $0$
Constant: $-11$

When a power of the variable is missing from a polynomial, its coefficient is implicitly zero.

Worked Example 5Constant term (NCERT Q5)

NCERT

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

The constant is the term without any variable. Here it is $-10$ .

Cross-check: the constant term is always equal to the value of the polynomial at $x = 0$ . Substituting:

$9(0)^3 + 5(0)^2 - 8(0) - 10 = -10. ✓$

This is a handy mental trick — to find the constant term of any polynomial, just plug in $x = 0$ .

Loading simulator…

The constant-difference signature of every linear pattern

The rule. For any linear polynomial $f(n) = an + b$ :

$f(n+1) - f(n) = [a(n+1) + b] - [an + b] = a$

The successive differences are always equal to the leading coefficient $a$ . Always. No matter where you are on the line.

This gives a test for linearity:

Compute $f(1), f(2), f(3), f(4), \ldots$
Look at the differences $f(2) - f(1)$ , $f(3) - f(2)$ , etc.
If all differences are equal → the pattern is linear.
If the differences themselves change (like for $f(n) = n^2$ , where they go $3, 5, 7, 9, \ldots$ ) → not linear.

The simulator above lets you toggle a comparison with $n^2$ to see this difference visually.

Logical ReasoningLevel 3 · Analysis

Look at the sequence $1, 4, 9, 16, 25, 36, \ldots$. Each number is the square of its position: $1 = 1^2$, $4 = 2^2$, etc. **Is this a linear pattern? Apply the constant-difference test.**

Quick Check

Q1.What is the leading coefficient $a$ in the linear polynomial $f(m) = 50m + 200$ ?