Introduction to Linear Polynomials

Contents hide

1Algebraic Expressions and Polynomials

2Linear Polynomials and Linear Patterns

3Finding the Relationship y = ax + b

4Graphing Lines: Slope and y-intercept

Chapter mind map: how it all connects

1 · Algebraic expressions and polynomialsTerms, variables, coefficients, and degree decide what kind of polynomial an expression is

2 · Linear polynomials and patternsDegree 1 means a constant difference between consecutive values: linear growth or linear decay

Introduction to Linear Polynomials

3 · Finding the relationship y = ax + bTwo data points are enough to pin down any linear relationship

4 · Graphing lines: slope and y-intercepta decides the steepness and direction, b decides where the line crosses the y-axis

What you will learn in this chapter

How to identify the terms, variables, coefficients and degree of a polynomial
What makes a pattern “linear”, and the difference between linear growth and linear decay
How to find the relationship y = ax + b between two variables from real data
How to plot a line and read off its slope and y-intercept, and what makes two lines parallel

algebraic expressionpolynomialdegreelinear polynomiallinear growthlinear decayslopey-interceptparallel lines

1Algebraic Expressions and Polynomials

A shop sells sealed boxes: red boxes have 4 pens each, blue boxes have 5 pencils each. If Meera buys x red boxes and y blue boxes, and gets 3 extra pens free, the total number of pens and pencils is $4x + 5y + 3$ . This is an algebraic expression: a combination of numbers, variables and operation symbols.

Learn by heartDefinition 1

In an algebraic expression, the parts separated by + or − signs are its terms. The letters standing for unknown numbers are variables. The number multiplying a variable is its coefficient, and a term with no variable is the constant term.

In $4x + 5y + 3$ : the terms are $4x$ , $5y$ and $3$ ; the variables are $x$ and $y$ ; the coefficients of $x$ and $y$ are $4$ and $5$ ; and $3$ is the constant term.

One variable vs two variables

This chapter deals only with expressions in one variable, such as $x^2+5x+1$ or $3z+7$ . An expression like $4x+5y+3$ , with two variables, is not the kind of polynomial studied here.

Learn by heartDefinition 2

An algebraic expression in one variable and its powers is called a polynomial (or univariate polynomial). The highest power of the variable in a polynomial is its degree.

Degree	Name	Example
0	Constant polynomial	$8$ (written $8x^0$ )
1	Linear polynomial	$3z + 7$
2	Quadratic polynomial	$x^2 + 5x + 1$
3	Cubic polynomial	$5y^3 + y^2 + 2y – 1$

Solved Example

Q. Find the degree and coefficient of $x^2$ in $7x^3 – 2x^2 + 5x – 9$ .

A. Highest power of $x$ is 3, so degree $= 3$ (a cubic polynomial). Coefficient of $x^2$ is $\;$ $-2$

2Linear Polynomials and Linear Patterns

Learn by heartDefinition 3

A polynomial of degree 1, such as $4x$ or $200 + 50m$ , is called a linear polynomial. When a linear polynomial is equated to a constant, we get a linear equation.

A chess club charges a joining fee of ₹200 plus ₹50 for every match played. If $m$ is the number of matches, the total cost is $200 + 50m$ , a linear polynomial in $m$ .

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

Learn by heartDefinition 4

A linear pattern is a sequence of numbers in which the difference between any two consecutive terms is constant. When that constant difference is added each time, we call it linear growth; when it is subtracted, linear decay.

Linear growth

Quantity increases by a fixed amount
Chess fee: +₹50 per match
Represented by a line with positive slope

Linear decay

Quantity decreases by a fixed amount
Pocket money: −₹5 per day
Represented by a line with negative slope

Solved Example

Q. A growing pattern of square tiles has 1, 3, 5, 7, 9 … tiles at Stages 1, 2, 3, 4, 5. Find the rule for the number of tiles at Stage $n$ , and the number of tiles at Stage 20.

The difference between consecutive terms is constantly 2, so this is a linear pattern. Each stage has one less than twice the stage number: tiles $= 2n – 1$ .

At Stage 20: $2(20) – 1 = \;$ $39$ tiles

Common Mistake

Not every increasing pattern is linear. A pattern is linear only if the difference between consecutive terms stays exactly the same every time, not just “getting bigger”.

3Finding the Relationship y = ax + b

A linear relationship between two variables $x$ and $y$ is written $y = ax + b$ . If we know two pairs of values, we can find $a$ and $b$ by solving two equations together.

Given

A data plan bill is $y = ax+b$ , where $x$ = GB used. 10 GB costs ₹350; 20 GB costs ₹550

Find

The values of $a$ and $b$

Solution

Substituting the two data points: $350 = 10a + b$ and $550 = 20a + b$

From the first equation, $b = 350 – 10a$ . Substituting into the second: $550 = 20a + 350 – 10a \Rightarrow 10a = 200 \Rightarrow a = 20$

Then $b = 350 – 10(20) = 150$ . So the relationship is $\;$ $y = 20x + 150$

Exam Tip

Here $a = 20$ means each extra GB costs ₹20, and $b = 150$ means there is a fixed monthly charge of ₹150 even with 0 GB used. Reading meaning into $a$ and $b$ is a common follow-up question.

4Graphing Lines: Slope and y-intercept

To plot $y = 2x + 1$ , we only need two points. When $x=0$ , $y=1$ , giving point A $(0,1)$ . When $x=3$ , $y=7$ , giving point B $(3,7)$ . Plotting A and B and joining them with a ruler, extended in both directions, gives the line.

**Figure 1**: The line y = 2x + 1, drawn from just two points A(0, 1) and B(3, 7).

Checking a point lies on a line

A point lies on a line exactly when its coordinates satisfy the line’s equation. For $y=2x+1$ : does $(7,15)$ lie on it? Substitute $x=7$ : $2(7)+1=15$ , which matches, so yes.

Plotting families of lines through the origin shows what the number $a$ in $y=ax$ controls:

**Figure 2**: y = 2x, y = x and y = x/2 all pass through the origin. A larger a makes the line steeper.

Learn by heartDefinition 5

In $y=ax+b$ , $a$ is called the slope of the line. It measures both direction (positive slope rises, negative slope falls) and steepness (larger $|a|$ is steeper).

**Figure 3**: y = –3x, y = –x and y = –x/3 all fall from left to right; the negative slope means linear decay.

Now compare lines with the same slope but a different constant term:

**Figure 4**: y = 2x – 1, y = 2x + 1 and y = 2x + 5 all have slope 2. Changing only b shifts the line up or down without changing its direction.

**Figure 5**: Each line crosses the y-axis at a different point: (0, 3), (0, 5) and (0, –2).

Learn by heartDefinition 6

In $y=ax+b$ , the constant $b$ is the y-intercept: the line always crosses the y-axis at the point $(0, b)$ .

Formula

y = ax + b

a

= slope of the line ·

b

= y-intercept, the point

(0,b)

where the line meets the y-axis

✓ True

Keeping $b$ fixed and changing $a$ changes the slope, not the y-intercept
Keeping $a$ fixed and changing $b$ shifts the line up or down, keeping it parallel to the original
Lines $y=ax+b$ with the same $a$ but different $b$ are parallel

✗ False

Changing $b$ makes the line steeper or flatter
Two lines with different slopes are parallel if their y-intercepts are close enough
In $y=ax$ there is no constant term, so $a$ must be 0

Did you know?

An auto-rickshaw fare that starts at ₹25 and adds ₹15 per km beyond the first 2 km is a real linear relationship: fare $= 15n – 5$ for $n \ge 2$ km. The slope, ₹15, is exactly the per-km rate; you will meet the same idea again when you study speed as the slope of a distance-time graph.

Every straight line is completely described by just two numbers: its slope and its y-intercept.

the single idea to carry out of this chapter

Check yourself before the exam

Can I find the degree, coefficients and constant term of a polynomial on sight?
Can I tell linear growth from linear decay just from the sign of the slope?
Can I find a and b in y = ax + b from two given data points?
Can I plot a line from two points, and explain what its slope and y-intercept mean?

Quick Revision: read this the night before the exam

Degree of a polynomial = highest power of the variable; degree 1 is linear
Linear pattern: constant difference between consecutive terms (growth = added, decay = subtracted)
y = ax + b: find a, b by substituting two known (x, y) pairs and solving
Any two points fix a line; a point lies on a line iff it satisfies the equation
a = slope (steepness and direction); b = y-intercept, the point (0, b)
Same a, different b: parallel lines. Same b, different a: lines crossing the y-axis at the same point

Practice Questions

1 markWrite a polynomial of degree 3 in x, where the coefficient of $x^2$ is $-4$ .
1 markFind the value of $4t^3 – t^2 + 6$ when $t = 2$ .
2 marksIf a number is multiplied by 3 and 5 is added, the result is $-16$ . Find the number.
2 marksA positive number is 4 times another number. If 20 is added to both, one new number becomes twice the other. Find the numbers.
2 marksRavi has ₹600 and saves ₹200 every month. Write a linear expression for the amount after $n$ months, and find the amount after 8 months.
3 marksThe digits of a two-digit number differ by 2. Reversing the digits and adding to the original number gives 132. Find the number.
4 marksDraw the graphs of $y = -2x+3$ and $2y = 4x+7$ (rewrite in $y=ax+b$ form first). Find their slopes and y-intercepts. Are they parallel?
3 marksThe graph of a linear polynomial $p(x)$ passes through $(2, 7)$ and $(4, 13)$ . Find $p(x)$ .
4 marksLet $p(x) = ax+b$ and $q(x) = cx+d$ . Given $p(0)=4$ , $p(x)-q(x)$ cuts the x-axis at $(1,0)$ , and $p(x)+q(x) = 8x+6$ for all $x$ . Find $p(x)$ and $q(x)$ .
3 marksWhat do all lines of the form $f(x) = ax + a$ , $a > 0$ , have in common? Explain using the y-intercept.

Multiple Choice

1 markThe degree of the polynomial $5 – 3x^2 + x^4$ is:
(a) 5(b) 4(c) 3(d) 2
1 markThe y-intercept of the line $3y = 6x – 9$ is:
(a) 6(b) 2(c) –3(d) –9
1 markWhich pair of lines is parallel?
(a) y = 3x+1, y = 3x-4(b) y = 2x, y = -2x(c) y = x+2, y = 2x+2(d) y = 5, x = 5

Assertion (A): The lines

y = 4x + 3

and

y = 4x – 7

never meet.
Reason (R): Lines with the same slope and different y-intercepts are parallel.

Case Study

A water tank has 500 litres at the start. A pump adds 40 litres every minute. Let

y

be the litres in the tank after

x

minutes.

1 markWrite the linear relationship between $y$ and $x$ .
1 markIs this linear growth or linear decay? What is the slope?
2 marksHow many litres are in the tank after 12 minutes?

Answer Key

Q1e.g.

x^3 – 4x^2 + x + 1

(any degree-3 polynomial with

x^2

coefficient

-4

)

4(8) – 4 + 6 = 34

3x+5=-16 \Rightarrow x = -7

Q4Numbers are 10 and 40 (since

4x+20 = 2(x+20) \Rightarrow x=10

)

Q5Amount

= 600+200n

; after 8 months

= ₹2200

Q6The number is 57 (or 75); digits differ by 2 and

57+75=132

y=-2x+3

: slope

-2

, y-intercept 3.

2y=4x+7 \Rightarrow y=2x+3.5

: slope 2, y-intercept 3.5. Not parallel (different slopes)

p(x) = 3x+1

p(x)=3x+4

q(x)=5x+2

Q10All pass through

(0,a)

, i.e. the y-intercept always equals the slope

MCQ 1–3(b) · (c) · (a)

A/RBoth A and R are true, and R is the correct explanation of A

Case Study

y = 40x+500

· linear growth, slope 40 · 980 litres