Class 9 › Mathematics › Chapter

Introduction to Linear Polynomials

Polynomials and their degree, linear polynomials, linear equations, linear patterns, growth and decay, the relationship $y = ax + b$ and its straight-line graph (slope and y-intercept).

1. Algebraic Expressions

Algebraic expression: a combination of numbers, variables and operation symbols ($+, -, \times, \div$).

Variable

A letter used for a quantity that can take different values (also called a "letter-number").

Term

Each part of an expression that is added or subtracted.

Coefficient

The number multiplying the variable part of a term (with its sign).

Constant term

A term with no variable; its value never changes.

Expressions can involve one variable or two or more variables; a term can be a product of variables.
The sign belongs to the term: in $2x^2 + 5xy - 3y^2$ the coefficients are $2$, $5$ and $-3$.
This chapter is restricted to expressions in one variable.

2. Polynomials in One Variable

One-variable (univariate) polynomial: an algebraic expression involving one variable and its powers. Often just called a polynomial when the context is clear. ("Univariate" means "having one variable".)

General form (variable $x$) $$p(x) = a_n x^n + a_{n-1}x^{n-1} + \dots + a_1 x + a_0$$

$a_n, \dots, a_1$ are the coefficients of $x^n, \dots, x$; $a_0$ is the constant term.
The powers of the variable in the terms are non-negative whole numbers.
The coefficient of a power that does not appear is $0$.
Any letter can be the variable ($x$, $y$, $z$, $t$, $n$, ...).

3. Degree and Types of Polynomials

Degree: the highest power of the variable in the polynomial.

Degree	Name	General form
0	Constant polynomial	$c = c\,x^0$ (a non-zero number)
1	Linear polynomial	$ax + b$, with $a \neq 0$
2	Quadratic polynomial	$ax^2 + bx + c$, with $a \neq 0$
3	Cubic polynomial	$ax^3 + bx^2 + cx + d$, with $a \neq 0$

A constant is a polynomial of degree $0$ because it can be written with $x^0$ (and $x^0 = 1$).
Degree is decided by the highest power that actually appears (with a non-zero coefficient), not by the number of terms or the order they are written in.
To find the coefficient of a particular power, read the number multiplying that power, with its sign.

4. Value of a Polynomial and the Input–Output (Function) Idea

Value of a polynomial

To evaluate a polynomial $p(x)$, substitute the given value of the variable in place of every $x$ and simplify. The result is written $p(a)$ for $x = a$.
Use brackets when substituting negative values (for example, put $-6$ in brackets so that signs and powers are handled correctly).

Polynomial as an input–output process: for every input value of $x$ there is exactly one output value of the polynomial. A rule that gives one output for each input is called a function, and we say the polynomial is a function of $x$.
• A linear polynomial gives a linear function; a quadratic polynomial gives a quadratic function.
• Written as $y = p(x)$ or $f(x)$: $x$ is the input (independent variable) and $y$ is the output (dependent variable).

5. Linear Polynomials

Linear polynomial: a polynomial of degree 1 in one variable, of the form $ax + b$ where $a \neq 0$.

$a$ is the coefficient of the variable; $b$ is the constant term (it may be $0$).
Key feature: when the input increases by $1$, the output always changes by the same fixed amount, namely $a$. The difference between successive values at integers is constant.

Why the difference is constant

1. For $p(x) = ax + b$, $p(x+1) = a(x+1) + b = ax + a + b$.

2. So $p(x+1) - p(x) = a$, whatever the value of $x$.

3. If the input increases by a step $h$, the output changes by $ah$ each time.

6. Linear Equations

Linear equation: obtained by equating a linear polynomial in one variable to a constant.

Linear equation in one variable and its solution $$ax + b = c \;\Longrightarrow\; x = \frac{c - b}{a}, \quad a \neq 0$$

Procedure for forming and solving a word problem

Step 1: let the unknown quantity be a variable (say $x$).

Step 2: express the other quantities in terms of $x$.

Step 3: use the given condition to form a linear polynomial and equate it to the given constant.

Step 4: solve for $x$ and then find every required quantity.

Step 5: check the answer against all the conditions in the problem.

Translation

Common phrases

"$n$ more than $x$" is $x + n$; "$n$ times $x$" is $nx$; "difference of $x$ and $y$" is $x - y$; "after $t$ years" adds $t$ to every present age.

Ratio

Ratio $m : n$

Take the two quantities as $mk$ and $nk$ for one unknown $k$.

Money & counts

Totals

Total value = (value of one item) $\times$ (number of items), added over all types of items.

Geometry

Perimeter

Express the sides in $x$ and use perimeter = sum of all sides (rectangle: $2(l + w)$).

7. Linear Patterns

Linear pattern: a sequence of numbers in which the difference between two consecutive terms is constant. This constant is the common difference $d$.

General (nth) term of a linear pattern $$t_n = t_1 + (n-1)d = dn + (t_1 - d)$$

The $n$th term is a linear polynomial in $n$ with degree $1$, so the relation between term number and term value is a linear relationship.
The coefficient of $n$ is the common difference $d$; the constant term is $t_1 - d$.

Steps to find the rule of a pattern: (1) write the values in a table, (2) find the constant difference $d$, (3) write $t_n = dn + c$ and find $c$ using the first term, (4) check with another term.
To find a term for a given $n$, substitute $n$. To find which term equals a given value, put $t_n$ equal to that value and solve the linear equation (the answer must be a positive whole number to be a term of the pattern).
Linear patterns are studied further in the chapter on Sequences and Progressions.

8. Linear Growth and Linear Decay

Linear growth

A quantity increases by a fixed amount over equal intervals. Slope is positive.

Linear decay

A quantity decreases by a fixed amount over equal intervals. Slope is negative.

Model for both $$f(t) = b + at$$

$b$ = initial value (value at $t = 0$).
$a$ = fixed change per unit interval (rate of change). $a \gt 0$ gives growth; $a \lt 0$ gives decay.
Value after $n$ intervals: $b + an$. To find when a target value $V$ is reached, solve $b + at = V$.
A decaying quantity such as a balance or amount left becomes zero when $b + at = 0$, i.e. $t = -\dfrac{b}{a}$.

Fixed charge plus rate: any quantity made of a fixed part plus a constant rate per unit is linear: $\text{total} = \text{fixed part} + (\text{rate}) \times (\text{units})$.

9. Linear Relationship $y = ax + b$

Linear relationship: a relationship between two variables $x$ and $y$ that can be written as $y = ax + b$ ($a$, $b$ constants).

$a$ (coefficient of $x$)

How much $y$ changes when $x$ increases by $1$: the rate or slope.

$b$ (constant term)

The value of $y$ when $x = 0$: the fixed / initial value or y-intercept.

Finding $a$ and $b$ from two known pairs of values $(x_1, y_1)$ and $(x_2, y_2)$

1. Substitute both pairs in $y = ax + b$: $\;y_1 = ax_1 + b$ and $y_2 = ax_2 + b$.

2. From the first, $b = y_1 - a x_1$. Substitute this in the second equation to eliminate $b$ and solve for $a$.

3. Equivalent result by subtracting the two equations: $\;a = \dfrac{y_2 - y_1}{x_2 - x_1}$ (with $x_1 \neq x_2$).

4. Then $b = y_1 - a x_1$. Write the relationship as $y = ax + b$.

Two-point result $$a = \frac{y_2 - y_1}{x_2 - x_1}, \qquad b = y_1 - a x_1$$

The same method gives a linear polynomial $p(x)$ that passes through two given points, and conversion rules such as between temperature scales.

10. Graphing a Linear Equation

The graph of $y = ax + b$ is a straight line on the coordinate plane. This is why such equations are called linear.

Method to plot $y = ax + b$

Step 1: choose any two values of $x$ and find the matching $y$ values (convenient choices: $x = 0$ gives the point $(0, b)$).

Step 2: plot the two points on the coordinate plane.

Step 3: join them with a ruler and extend the line in both directions (arrow heads).

Step 4 (check): find a third point; it must lie on the same line.

Point on a line: a point lies on the line if and only if its coordinates satisfy the equation (substitute $x$ and $y$ and check both sides are equal).
Finding an equation from points: if several points lie on one straight line, look for the rule connecting each $y$-coordinate with its $x$-coordinate (for instance, if every $y$ equals $k$ times $x$, the line is $y = kx$).
Where the line meets the axes: y-axis at $(0, b)$ (put $x = 0$); x-axis at $\left(-\dfrac{b}{a},\, 0\right)$ for $a \neq 0$ (put $y = 0$).
Reading $a$ and $b$ from other forms: rewrite the equation as $y = ax + b$ by dividing by the coefficient of $y$, then read off $a$ and $b$.

11. The Line $y = ax$ (Role of $a$)

Lines of the form $y = ax$ for different values of $a$.

Every line $y = ax$ passes through the origin $(0, 0)$, since $b = 0$.
$a$ is the slope of the line $y = ax$ (the idea of slope is developed in the chapter on linear equations).
$a \gt 0$: line passes through Quadrants I and III and rises to the right. $a \lt 0$: line passes through Quadrants II and IV and falls to the right.
Steepness depends on the size of $a$ (ignoring sign): $|a| \gt 1$ steeper than $y = x$, $|a| \lt 1$ less steep. $y = x$ and $y = -x$ are equally inclined to both axes.
The lines $y = ax$ and $y = -ax$ are mirror images of each other in an axis.

12. Slope and y-intercept of $y = ax + b$ (Role of $b$)

Lines with the same slope $a$ and different y-intercepts $b$.

Slope–intercept form $$y = ax + b \qquad (a = \text{slope},\; b = \text{y-intercept})$$

The line cuts the y-axis at the point $(0, b)$. $b$ is the signed distance of this point from the origin: $b \gt 0$ above the origin, $b \lt 0$ below it, $b = 0$ at the origin.

Change $a$, keep $b$

Slope changes

The y-intercept stays fixed, so all such lines pass through the same point $(0, b)$ but with different steepness.

Change $b$, keep $a$

Line shifts

The line moves up or down but stays parallel to the original line.

Parallel lines

Equal slopes

Lines with the same slope but different y-intercepts are parallel; they never meet.

Growth & decay

Sign of slope

Positive slope: line rises (linear growth). Negative slope: line falls (linear decay).

Patterns

Slope = difference

For a linear pattern, the slope of the line (term number vs term value) equals the constant difference between consecutive terms.

Compare

$y = ax + b$ vs $y = -ax + b$

Same y-intercept $(0, b)$, but one rises and the other falls: the slopes are equal in size and opposite in sign.

13. Quick Revision Sheet

Concept	Key points
Polynomial (one variable)	Variable with powers; degree = highest power
Degree 0 / 1 / 2 / 3	Constant / Linear / Quadratic / Cubic
Linear polynomial	$ax + b$, $a \neq 0$; output changes by $a$ per unit step in $x$
Linear equation	$ax + b = c \Rightarrow x = \dfrac{c - b}{a}$
Linear pattern	Constant difference $d$; $t_n = t_1 + (n-1)d$
Linear growth / decay	$f(t) = b + at$; $a \gt 0$ growth, $a \lt 0$ decay
Linear relationship	$y = ax + b$; slope $a$, y-intercept $b$
From two points	$a = \dfrac{y_2 - y_1}{x_2 - x_1}$, $b = y_1 - ax_1$
Graph	Straight line through $(0, b)$ and $\left(-\dfrac{b}{a}, 0\right)$
$y = ax$	Passes through origin; steeper as $\|a\|$ increases
Parallel lines	Same $a$, different $b$

Common mistakes:
• Taking the number of terms (instead of the highest power) as the degree.
• Dropping the sign of a coefficient (the coefficient of $x$ in $5 - 4x$ is $-4$).
• Forgetting brackets when substituting negative values.
• Mixing up $a$ (slope) and $b$ (y-intercept) when reading $y = ax + b$.
• Not rearranging an equation like $2y = 4x + 7$ into $y = ax + b$ before reading the slope and intercept.