Find the Common Difference and the Formula Writes Itself
Learn to identify a linear polynomial and state its degree, evaluate it and find its zero, turn a growing pattern into an expression in n, and read the rate of change off a table of values.
How do you find the formula for a growing pattern?
Look at the difference between consecutive stages. That difference is the coefficient, and the rest follows in one line.
Make a row of squares out of matchsticks. One square needs sticks, two squares need , three need , four need .
The differences are , , — the same every time. So each new square adds sticks, and the expression must begin .
Now fix the rest. At the expression gives , and the true answer is — so the constant is :
Check it at another stage. At : . Correct. At it predicts sticks, without anybody building the row.
That is a linear polynomial — an expression of the form — and this page is about where such expressions come from and what can be done with them. It covers the first part of the CBSE Class 9 Mathematics chapter on linear polynomials.
Make a row of squares out of matchsticks. One square needs sticks, two squares need , three need , four need .
The differences are , , — the same every time. So each new square adds sticks, and the expression must begin .
Now fix the rest. At the expression gives , and the true answer is — so the constant is :
Check it at another stage. At : . Correct. At it predicts sticks, without anybody building the row.
That is a linear polynomial — an expression of the form — and this page is about where such expressions come from and what can be done with them. It covers the first part of the CBSE Class 9 Mathematics chapter on linear polynomials.
Formula
What makes an expression a linear polynomial?
**Degree exactly **, which means it can be written as
Here is the **coefficient of ** and is the constant term.
What counts as a polynomial at all. In a polynomial in , every power of must be a non-negative whole number. So , and are allowed, and a constant is allowed as , but a negative or fractional power is not.
The degree is the highest power of that appears.
Worked classification.
- — linear. Coefficient of is , constant term , degree
- — linear, with and
- — linear, with and
- — linear, with
- — a constant polynomial of degree . Not linear
- — degree , a quadratic. Not linear
- — not a polynomial at all, because has a negative exponent
- — not a polynomial, because the exponent is
**Why is required.** If were the expression would collapse to alone — a constant of degree , whose graph is a horizontal line rather than a slanting one. So the condition is not a technicality; it is what makes the expression describe something that changes.
A fractional coefficient is fine; a fractional power is not. is a perfectly ordinary linear polynomial with , while is not a polynomial at all. The restriction is on the exponents, never on the coefficients — and that is the distinction the two examples above are chosen to separate.
One more boundary case. The degree of the zero polynomial is not defined, because there is no highest power to point to.
Here is the **coefficient of ** and is the constant term.
What counts as a polynomial at all. In a polynomial in , every power of must be a non-negative whole number. So , and are allowed, and a constant is allowed as , but a negative or fractional power is not.
The degree is the highest power of that appears.
Worked classification.
- — linear. Coefficient of is , constant term , degree
- — linear, with and
- — linear, with and
- — linear, with
- — a constant polynomial of degree . Not linear
- — degree , a quadratic. Not linear
- — not a polynomial at all, because has a negative exponent
- — not a polynomial, because the exponent is
**Why is required.** If were the expression would collapse to alone — a constant of degree , whose graph is a horizontal line rather than a slanting one. So the condition is not a technicality; it is what makes the expression describe something that changes.
A fractional coefficient is fine; a fractional power is not. is a perfectly ordinary linear polynomial with , while is not a polynomial at all. The restriction is on the exponents, never on the coefficients — and that is the distinction the two examples above are chosen to separate.
One more boundary case. The degree of the zero polynomial is not defined, because there is no highest power to point to.
How do you evaluate a linear polynomial and find its zero?
**To evaluate, substitute the value of . To find the zero, set the polynomial equal to and solve.
Worked example 1 — evaluating.** Let .
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**Worked example 2 — finding for a stated value.** For what does ?
Check: . Correct.
Worked example 3 — the zero of the polynomial. The zero is the value of making the polynomial equal to .
Worked example 4 — a second polynomial. Let .
- , ,
- Its zero: gives
The general result. For :
A linear polynomial has exactly one zero. There is one value of that makes it zero, never two and never none — which is a direct consequence of , since the equation can always be divided by .
**Notice that always.** Substituting leaves only the constant term, so the constant can be read off as the value at zero. That becomes useful in the next part of this chapter, where is the starting value of a real situation.
The value of the polynomial and the zero of the polynomial are different questions. asks *what comes out when goes in. The zero asks what must go in for to come out*. One is a substitution and the other is an equation to solve, and they are asked in almost identical words — so read carefully which is wanted.
Worked example 1 — evaluating.** Let .
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**Worked example 2 — finding for a stated value.** For what does ?
Check: . Correct.
Worked example 3 — the zero of the polynomial. The zero is the value of making the polynomial equal to .
Worked example 4 — a second polynomial. Let .
- , ,
- Its zero: gives
The general result. For :
A linear polynomial has exactly one zero. There is one value of that makes it zero, never two and never none — which is a direct consequence of , since the equation can always be divided by .
**Notice that always.** Substituting leaves only the constant term, so the constant can be read off as the value at zero. That becomes useful in the next part of this chapter, where is the starting value of a real situation.
The value of the polynomial and the zero of the polynomial are different questions. asks *what comes out when goes in. The zero asks what must go in for to come out*. One is a substitution and the other is an equation to solve, and they are asked in almost identical words — so read carefully which is wanted.
How do you write the expression for the nth stage of a pattern?
**The common difference gives , and the first term then gives ** by the rule .
That is the whole method, and it takes two lines once the differences are written down.
Worked example 1 — matchstick squares. A row of squares: , , , sticks.
- Differences: , , — so
- First term is , so
Check at two stages: gives , and gives . Both correct, and predicts .
Worked example 2 — matchstick triangles. A row of triangles: , , sticks.
- Differences: — so
- First term is , so
Check: gives ; gives .
Worked example 3 — a dot pattern. Dots counted at each stage: , , , .
- Differences: — so
- First term is , so
Check: gives ; predicts .
Worked example 4 — chairs at tables. One square table seats people. Each extra table pushed against it in a row adds seating for more.
- Differences: — so
- First term is , so
Check: gives ; predicts seats.
Check the differences are constant before assuming a linear expression fits. Take the pattern , , , . Its differences are , , — not constant. No expression of the form can produce it, and the correct expression is , which is quadratic.
So the first thing to do with any pattern is to write down the differences. Constant differences mean a linear expression exists and the method above will find it. Varying differences mean it does not, and looking for one is wasted effort.
Verify at more than one stage. Fitting the first term alone proves nothing, because was chosen to make the first term work. The second and third stages are the real test — and checking one of them takes five seconds.
That is the whole method, and it takes two lines once the differences are written down.
Worked example 1 — matchstick squares. A row of squares: , , , sticks.
- Differences: , , — so
- First term is , so
Check at two stages: gives , and gives . Both correct, and predicts .
Worked example 2 — matchstick triangles. A row of triangles: , , sticks.
- Differences: — so
- First term is , so
Check: gives ; gives .
Worked example 3 — a dot pattern. Dots counted at each stage: , , , .
- Differences: — so
- First term is , so
Check: gives ; predicts .
Worked example 4 — chairs at tables. One square table seats people. Each extra table pushed against it in a row adds seating for more.
- Differences: — so
- First term is , so
Check: gives ; predicts seats.
Check the differences are constant before assuming a linear expression fits. Take the pattern , , , . Its differences are , , — not constant. No expression of the form can produce it, and the correct expression is , which is quadratic.
So the first thing to do with any pattern is to write down the differences. Constant differences mean a linear expression exists and the method above will find it. Varying differences mean it does not, and looking for one is wasted effort.
Verify at more than one stage. Fitting the first term alone proves nothing, because was chosen to make the first term work. The second and third stages are the real test — and checking one of them takes five seconds.
What does a table of values tell you about the rate of change?
**The change in for each unit increase in is exactly the coefficient of ** — and its sign says whether rises or falls.
Worked example 1 — a rising polynomial. Build a table for .
- : , , , ,
- : , , , ,
The differences in are , , , . So each unit increase in raises by ****, which is the coefficient of .
Worked example 2 — a falling polynomial. Build a table for .
- : , , ,
- : , , ,
The differences are each time. So each unit increase in lowers by — again the coefficient, and the minus sign is doing the work.
Worked example 3 — reading the polynomial off a table. Given
- : , , ,
- : , , ,
The differences are each time, so . And at the value is , so :
**Check at **: . Correct.
Worked example 4 — where the table does not start at zero. Given
- : , , ,
- : , , ,
Differences are , so . Now must be found from any one row rather than read off, since is not listed. Using , :
**Check at **: . Correct.
Constant differences are exactly what makes a relationship linear. That is the definition, restated in the language of a table. If the differences vary, no expression will fit — and the pattern , , , from the previous section is the standard example of a table that fails the test.
**Read as the value at whenever the table provides it. When it does not, substitute any one row and solve, as worked example 4 did. Do not assume the first listed is ** — in that example the first listed value was and was , and taking the first value as the constant is the standard error when a table starts somewhere other than zero.
The next part of this chapter takes these same tables and plots them, and the constant difference shows up there as the steepness of a straight line.
Worked example 1 — a rising polynomial. Build a table for .
- : , , , ,
- : , , , ,
The differences in are , , , . So each unit increase in raises by ****, which is the coefficient of .
Worked example 2 — a falling polynomial. Build a table for .
- : , , ,
- : , , ,
The differences are each time. So each unit increase in lowers by — again the coefficient, and the minus sign is doing the work.
Worked example 3 — reading the polynomial off a table. Given
- : , , ,
- : , , ,
The differences are each time, so . And at the value is , so :
**Check at **: . Correct.
Worked example 4 — where the table does not start at zero. Given
- : , , ,
- : , , ,
Differences are , so . Now must be found from any one row rather than read off, since is not listed. Using , :
**Check at **: . Correct.
Constant differences are exactly what makes a relationship linear. That is the definition, restated in the language of a table. If the differences vary, no expression will fit — and the pattern , , , from the previous section is the standard example of a table that fails the test.
**Read as the value at whenever the table provides it. When it does not, substitute any one row and solve, as worked example 4 did. Do not assume the first listed is ** — in that example the first listed value was and was , and taking the first value as the constant is the standard error when a table starts somewhere other than zero.
The next part of this chapter takes these same tables and plots them, and the constant difference shows up there as the steepness of a straight line.
Exam tip
Exam tip: write the differences first, then check at a second stage
Write the differences between consecutive terms before anything else. Constant differences mean a linear expression exists; varying differences mean it does not — has differences and is not linear.
**The common difference is **, and then . For : , , so .
Verify at a second stage. Fitting the first term proves nothing, because was chosen to make it fit.
For a linear polynomial, degree is exactly and . A constant such as has degree and is not linear.
The restriction is on the exponents, not the coefficients. is linear; and are not polynomials.
Distinguish the value from the zero. is a substitution; the zero solves , giving . A linear polynomial has exactly one zero.
** always**, so the constant is the value at .
In a table, the change in per unit increase in is the coefficient of , and its sign says rising or falling.
**Do not assume the first listed is .** If the table does not start at , substitute one row and solve — for with , is , not .
And state the coefficient, the constant and the degree separately when a question asks for all three.
**The common difference is **, and then . For : , , so .
Verify at a second stage. Fitting the first term proves nothing, because was chosen to make it fit.
For a linear polynomial, degree is exactly and . A constant such as has degree and is not linear.
The restriction is on the exponents, not the coefficients. is linear; and are not polynomials.
Distinguish the value from the zero. is a substitution; the zero solves , giving . A linear polynomial has exactly one zero.
** always**, so the constant is the value at .
In a table, the change in per unit increase in is the coefficient of , and its sign says rising or falling.
**Do not assume the first listed is .** If the table does not start at , substitute one row and solve — for with , is , not .
And state the coefficient, the constant and the degree separately when a question asks for all three.
Did you know
Why a pattern that fits three terms can still be wrong
Take the sequence , , . What comes next?
The obvious answer is — each term is double the one before. But is equally defensible, if the rule is add one, then add two, then add three. So is under a third rule, and other numbers under others.
Three terms do not settle a rule, and neither do four, or ten. Any finite list of numbers can be continued in more than one way, and each continuation comes with a formula that fits every term given.
That is why the method on this page is careful to say what it is doing: it finds the linear expression fitting a pattern if one exists, and the test for whether one exists is the constant difference. It does not prove that the pattern must continue linearly — only that a linear rule is consistent with what has been shown.
For a matchstick pattern that is enough, because the structure supplies the missing certainty. Each new square genuinely does need three more sticks, for a reason you can see by looking at the row: three new sticks and one shared with the square before. The formula is not a guess fitted to numbers; it is a count, and the numbers merely confirm it.
So a pattern question has two halves. Finding the expression is arithmetic, and the differences give it in one line. Justifying it means saying why the structure grows that way — three new sticks per square, or two new chairs per table.
An answer with the formula and the reason is complete. An answer with only the formula has fitted the numbers and hoped, which is exactly what the sequence shows to be risky.
The obvious answer is — each term is double the one before. But is equally defensible, if the rule is add one, then add two, then add three. So is under a third rule, and other numbers under others.
Three terms do not settle a rule, and neither do four, or ten. Any finite list of numbers can be continued in more than one way, and each continuation comes with a formula that fits every term given.
That is why the method on this page is careful to say what it is doing: it finds the linear expression fitting a pattern if one exists, and the test for whether one exists is the constant difference. It does not prove that the pattern must continue linearly — only that a linear rule is consistent with what has been shown.
For a matchstick pattern that is enough, because the structure supplies the missing certainty. Each new square genuinely does need three more sticks, for a reason you can see by looking at the row: three new sticks and one shared with the square before. The formula is not a guess fitted to numbers; it is a count, and the numbers merely confirm it.
So a pattern question has two halves. Finding the expression is arithmetic, and the differences give it in one line. Justifying it means saying why the structure grows that way — three new sticks per square, or two new chairs per table.
An answer with the formula and the reason is complete. An answer with only the formula has fitted the numbers and hoped, which is exactly what the sequence shows to be risky.
Exam relevance
How do linear polynomials feed into JEE Main?
Because polynomials are the objects almost all of algebra operates on, and the vocabulary fixed on this page is used without restatement for the next three years.
This is the foundation for the Class 9 and Class 10 chapters on Polynomials and then for Class 11 Mathematics Complex Numbers and Quadratic Equations and Sequences and Series, all examined in JEE Main. The definitions of degree, coefficient, constant term and zero established here carry over unchanged; what grows is the degree of the polynomials handled.
The zero of a polynomial is the idea with the longest reach. Class 10 extends it to quadratics, where there are two zeros and the relationships between the zeros and the coefficients appear. Class 11 extends it to complex zeros, and the factor theorem — that is a factor exactly when is a zero — becomes a standard tool. The Class 9 point that a linear polynomial has exactly one zero is the base case of a general result about degree and the number of zeros.
The constant-difference test becomes the arithmetic progression in Class 10 and Class 11 Sequences and Series. The expression found on this page is the th term of an AP with common difference , written there as — the same expression rearranged. A student who found from the differences has already done the essential step of that chapter.
What is not a polynomial matters more later than it appears to now. Expressions with negative or fractional exponents belong to rational and irrational functions, and Class 11 Relations and Functions treats their domains separately for exactly that reason.
What the questions look like. For board work, identify the polynomial and state its degree and find the zero are standard short questions, along with **write the th term of a given pattern. For JEE Main, this page appears only as assumed knowledge — the degree of an expression, the number of zeros expected, and the arithmetic progression from a constant difference are all steps inside harder problems.
How board and competitive emphasis differ. A board paper asks you to classify several expressions, evaluate a polynomial at given values, find its zero, and extend a pattern with its formula — each self-contained. A competitive paper never asks what a degree is; it asks something whose answer depends on knowing. So the board paper rewards the definitions and the competitive paper rewards having them automatic.
The single trap that costs the most marks. Confusing the value of a polynomial with its zero**. means substitute ; the zero means solve . The two are asked in almost identical language — *find at against find the zero of * — and reading the wrong one produces a perfectly correct answer to a question nobody asked.
A second trap worth naming. Treating a fractional coefficient as disqualifying. is a linear polynomial with ; is not a polynomial at all. The restriction is on the exponents, and a question listing both is testing exactly that distinction.
This is the foundation for the Class 9 and Class 10 chapters on Polynomials and then for Class 11 Mathematics Complex Numbers and Quadratic Equations and Sequences and Series, all examined in JEE Main. The definitions of degree, coefficient, constant term and zero established here carry over unchanged; what grows is the degree of the polynomials handled.
The zero of a polynomial is the idea with the longest reach. Class 10 extends it to quadratics, where there are two zeros and the relationships between the zeros and the coefficients appear. Class 11 extends it to complex zeros, and the factor theorem — that is a factor exactly when is a zero — becomes a standard tool. The Class 9 point that a linear polynomial has exactly one zero is the base case of a general result about degree and the number of zeros.
The constant-difference test becomes the arithmetic progression in Class 10 and Class 11 Sequences and Series. The expression found on this page is the th term of an AP with common difference , written there as — the same expression rearranged. A student who found from the differences has already done the essential step of that chapter.
What is not a polynomial matters more later than it appears to now. Expressions with negative or fractional exponents belong to rational and irrational functions, and Class 11 Relations and Functions treats their domains separately for exactly that reason.
What the questions look like. For board work, identify the polynomial and state its degree and find the zero are standard short questions, along with **write the th term of a given pattern. For JEE Main, this page appears only as assumed knowledge — the degree of an expression, the number of zeros expected, and the arithmetic progression from a constant difference are all steps inside harder problems.
How board and competitive emphasis differ. A board paper asks you to classify several expressions, evaluate a polynomial at given values, find its zero, and extend a pattern with its formula — each self-contained. A competitive paper never asks what a degree is; it asks something whose answer depends on knowing. So the board paper rewards the definitions and the competitive paper rewards having them automatic.
The single trap that costs the most marks. Confusing the value of a polynomial with its zero**. means substitute ; the zero means solve . The two are asked in almost identical language — *find at against find the zero of * — and reading the wrong one produces a perfectly correct answer to a question nobody asked.
A second trap worth naming. Treating a fractional coefficient as disqualifying. is a linear polynomial with ; is not a polynomial at all. The restriction is on the exponents, and a question listing both is testing exactly that distinction.
Key takeaways
Linear polynomials, zeros and patterns: quick revision
- A linear polynomial has degree exactly and is written with ****. Here is the **coefficient of ** and the constant term.
- In a polynomial every exponent must be a non-negative whole number, and the degree is the highest power present.
- Linear: , , (with ), (with ).
- Not linear: (degree , a constant), (quadratic).
- Not polynomials at all: (negative exponent), (fractional exponent).
- The restriction is on the exponents, not the coefficients — a fraction as a coefficient is fine.
- Evaluating : , , , .
- For a stated value: gives .
- The zero solves , so . For it is ; for it is .
- A linear polynomial has exactly one zero, and ** always.
- Value and zero are different questions — one substitutes, the other solves.
- For a pattern: the common difference is **, and .
- Matchstick squares give ; triangles give ; dots give ; chairs give .
- Check the differences are constant first. has differences , so no linear expression fits — it is .
- Verify at a second stage, since was chosen to fit the first.
- In a table, the change in per unit increase in is the coefficient. rises by each step; falls by .
- Reading a polynomial off a table: differences of with at gives .
- If the table does not start at , substitute a row: with gives , so — not .
- Justify a pattern from its structure as well as fitting the numbers — three new sticks per square, two new chairs per table.
Build a matchstick pattern of your own, write its expression from the differences, then predict the tenth stage and build it to check — a prediction you have verified yourself is worth more than one you were given.
- In a polynomial every exponent must be a non-negative whole number, and the degree is the highest power present.
- Linear: , , (with ), (with ).
- Not linear: (degree , a constant), (quadratic).
- Not polynomials at all: (negative exponent), (fractional exponent).
- The restriction is on the exponents, not the coefficients — a fraction as a coefficient is fine.
- Evaluating : , , , .
- For a stated value: gives .
- The zero solves , so . For it is ; for it is .
- A linear polynomial has exactly one zero, and ** always.
- Value and zero are different questions — one substitutes, the other solves.
- For a pattern: the common difference is **, and .
- Matchstick squares give ; triangles give ; dots give ; chairs give .
- Check the differences are constant first. has differences , so no linear expression fits — it is .
- Verify at a second stage, since was chosen to fit the first.
- In a table, the change in per unit increase in is the coefficient. rises by each step; falls by .
- Reading a polynomial off a table: differences of with at gives .
- If the table does not start at , substitute a row: with gives , so — not .
- Justify a pattern from its structure as well as fitting the numbers — three new sticks per square, two new chairs per table.
Build a matchstick pattern of your own, write its expression from the differences, then predict the tenth stage and build it to check — a prediction you have verified yourself is worth more than one you were given.