Polynomials – Class 10 Maths Notes & Practice

Think of a polynomial as a machine that accepts a number, performs a fixed algebraic calculation, and gives back another number. In Class 10 Maths, the chapter becomes powerful when you stop seeing a polynomial only as a line of symbols and start asking two practical questions: what input makes it become zero, and what do the coefficients secretly tell us about those zero-making inputs?

This chapter keeps the focus exactly where a CBSE Class 10 student needs it: recognising polynomials, understanding zeros, reading zeros from simple graphs, and using the relationship between the zeros and coefficients of a quadratic polynomial. The explanations are deliberately step-by-step because most mistakes in this chapter are not caused by hard algebra; they are caused by sign slips, unclear vocabulary, or forgetting whether the question asks for the value of the polynomial or the value of x.

We will use original examples throughout. Treat each worked example like a mini-conversation: first identify what the question is really asking, then choose the quickest method, then check the answer with substitution or with the coefficient relationship. By the end, you should be able to solve standard school questions and also explain why your answer is correct.

Polynomial Basics: Terms, Powers and Degree

A polynomial in one variable is made from constants and non-negative whole-number powers of the variable. Expressions such as 4x^3 - 2x + 7, x^2 + 5, and 9 are polynomials. Expressions with negative powers, variables in denominators, or square roots of variables are not polynomials in the Class 10 sense. For example, 3/x + 2 is not a polynomial in x because it contains x^-1.

The degree of a non-zero polynomial is the highest power of the variable with a non-zero coefficient. A constant non-zero polynomial has degree zero. A zero polynomial is a special case because all its coefficients are zero, so its degree is not defined in the usual way. In school questions, when you see a polynomial like 5x^2 + 0x + 1, ignore the missing or zero coefficient term while deciding degree.

The common names come from degree. Degree one is linear, degree two is quadratic, and degree three is cubic. A linear polynomial can have at most one zero. A quadratic can have at most two real zeros. A cubic can have at most three real zeros. The phrase ‘at most’ matters because the actual number can be lower when the graph touches the axis once or misses it.

What a Zero of a Polynomial Really Means

A zero of a polynomial is a value of the variable that makes the polynomial equal to zero. If p(a) = 0, then a is a zero of p(x). The word zero refers to the output of the polynomial, but the answer you write is the input value that caused that output.

This is where many students get confused. Suppose p(x) = x - 4. The zero is 4, because p(4) = 0. We do not say the zero is 0, because zero is the result after substitution. The zero of the polynomial is the value of x.

For a linear polynomial, finding the zero usually means solving a simple equation. For a quadratic, it often means factorising and putting each factor equal to zero. You can also be asked only to check whether a proposed number is a zero; in that case, substitution is enough and factorisation is not necessary.

The Graph Picture: Zeros Are X-Axis Contacts

If you draw the graph of y = p(x), a zero of p(x) appears where the graph meets the x-axis. Why? Every point on the x-axis has y-coordinate zero. So when the polynomial value becomes zero, the graph has reached the x-axis.

A linear polynomial has a straight-line graph and can meet the x-axis at one point. A quadratic polynomial has a U-shaped or inverted-U-shaped graph called a parabola. It may cut the x-axis at two different points, touch it at one point, or avoid the x-axis entirely. In the first case there are two real zeros, in the second case there is one repeated real zero, and in the third case there is no real zero.

For Class 10 questions, you normally do not need to draw a perfect graph. You only need to understand the meaning of x-axis crossings. If the question tells you that the graph cuts the x-axis at -1 and 6, it has already given the zeros. From there, you can write factors and build a polynomial.

Relationship Between Zeros and Coefficients

For a quadratic polynomial ax^2 + bx + c, where a is not zero, the coefficients reveal two facts about the zeros. If the zeros are alpha and beta, then their sum is -b/a and their product is c/a. You should know these formulas, but you should also know where the signs come from.

Imagine the quadratic has zeros alpha and beta. Then a polynomial with those zeros can be written as k(x - alpha)(x - beta), where k is a non-zero constant. Expanding gives k[x^2 - (alpha + beta)x + alpha beta]. So the coefficient of x carries the negative of the sum, and the constant term carries the product. When we compare this with ax^2 + bx + c, we get alpha + beta = -b/a and alpha beta = c/a.

Use this relationship whenever the question gives a quadratic and asks for sum, product, verification, or an unknown coefficient. It is especially useful when the zeros are not easy to find separately. In many school questions, the examiner is not testing factorisation first; the examiner is testing whether you can read information directly from a, b, and c.

Building a Quadratic Polynomial from Given Zeros

Sometimes the direction reverses. Instead of giving the polynomial and asking for zeros, the question gives the zeros and asks you to form a polynomial. If the zeros are alpha and beta, a simple monic polynomial is x^2 - (alpha + beta)x + alpha beta. Monic means the coefficient of x^2 is one.

If fractions appear, you may multiply the whole polynomial by a non-zero constant to remove denominators. This does not change the zeros because multiplying by a non-zero number does not change where the expression becomes zero. For example, x^2 + (7/2)x - 2 and 2x^2 + 7x - 4 have the same zeros.

The phrase ‘a quadratic polynomial’ is important. There can be infinitely many polynomials with the same zeros because you can multiply by 2, 5, -1, or any non-zero constant. In school solutions, one clean polynomial is enough unless the question asks for a specific leading coefficient.

Exam Method: How to Choose the Right Move

The chapter becomes easy when you match the wording of the question with the correct method. Do not treat every question as a factorisation question. Sometimes the direct coefficient relationship is quicker, and sometimes a one-line substitution is all that is required.

Another useful habit is to write a, b, and c before using the quadratic relationship. If the polynomial is 2x^2 - 9x + 10, write a = 2, b = -9, and c = 10. This small line protects you from losing the negative sign attached to b.

When constructing a polynomial from zeros, spend a few seconds on the sum and product separately. Students who jump directly into the formula often make double-negative errors. If the zeros are -3 and 5, the sum is 2 and the product is -15; therefore the polynomial is x^2 - 2x - 15. Seeing the two intermediate values makes the final expression much easier to trust.

Deeper Tutor Notes for Stronger Answers

A zero is tied to an equation. When you find zeros of p(x), you are really solving p(x) = 0. This is why factorisation works: if a product of factors is zero, at least one factor must be zero. For example, if (x - 7)(x + 3) = 0, then either x - 7 = 0 or x + 3 = 0. That gives 7 and -3.

The coefficient relationship is not a separate trick floating outside the chapter. It comes directly from expanding factors. If the zeros are alpha and beta, the factors are (x - alpha) and (x - beta). Expanding gives x^2 - (alpha + beta)x + alpha beta. That is why the sum appears with a negative sign in the middle term and the product appears as the constant term.

The leading coefficient a matters because not every quadratic is monic. In 3x^2 - 12x + 9, the sum of zeros is not simply 12. It is -b/a = 12/3 = 4. The product is c/a = 9/3 = 3. Dividing by a normalises the polynomial before reading the relationship.

If a polynomial has two zeros with opposite signs, their product is negative. You can use this for mental checking. A polynomial like x^2 - 5x - 6 has constant term -6, so its two zeros, if real and integer, should have opposite signs. That matches -1 and 6.

If the constant term of a quadratic is positive and the middle coefficient is negative, two positive zeros are possible. For x^2 - 7x + 12, both zeros are positive because their product is positive and their sum is positive. This does not replace solving, but it helps you detect an answer such as -3 and -4, whose sum would be negative.

For repeated zeros, the sum and product relationship still works. In x^2 - 6x + 9, the zeros are 3 and 3. Sum is 6, product is 9. The graph touches the x-axis at one point, but algebra counts the zero twice because the factor (x - 3) appears twice.

When the question asks for ‘a polynomial’, avoid overthinking uniqueness. If zeros are 2 and 5, then x^2 - 7x + 10 is fine. So is 3x^2 - 21x + 30. Both become zero at the same two values. Unless a leading coefficient is specified, the monic form is the neatest answer.

Graph questions are usually concept questions, not drawing competitions. If a graph intersects the x-axis at two marked x-values, write those x-values as zeros. If the graph only touches the x-axis at one point, identify it as one repeated zero. If it stays above or below the x-axis, it has no real zero.

A quick substitution check is the best habit in this chapter. If your answer says 5/2 is a zero of 2x^2 - 9x + 10, calculate 2(25/4) - 9(5/2) + 10. That is 25/2 - 45/2 + 10 = -20/2 + 10 = 0. The check takes less than a minute.

Do not confuse coefficient with term. In -8x^2, the coefficient is -8, not 8. The sign travels with the coefficient. In the quadratic formula relationship used here, the sign of b is especially important because the sum uses -b/a.

If a polynomial is written out of order, first arrange it mentally in descending powers. For 5 - 2x + 3x^2, read it as 3x^2 - 2x + 5. Then a = 3, b = -2, and c = 5. The order on the page should not change the values of the coefficients.

In application-style algebra, polynomials often appear as area or number patterns. The zero then represents the value where the expression becomes zero, which may or may not make sense in the real situation. For Class 10, always solve the algebra first and then check whether the value fits the context, especially if a length or count is involved.

Mistake Clinic: The Errors That Cost Marks

The chapter is short, but it is full of sign-sensitive moves. Use this clinic as a checklist after solving practice questions. If an answer is wrong, it will usually belong to one of these patterns.

Question Method Bank: Read the Wording First

These method notes are meant for revision week. Before doing a question, identify which row it belongs to. That decision usually matters more than the arithmetic.

When the question says ‘verify’

Verification is not the same as discovery. If the question already gives possible zeros, do not spend time inventing another method first. Substitute each proposed zero into the polynomial and show that the value becomes zero. If the polynomial is quadratic and two proposed zeros are given, you may also verify their sum and product against -b/a and c/a. The best answer usually names the given numbers, performs the calculation neatly, and ends with a direct statement that the relationship or zero condition is verified.

When the question says ‘find the zeros’

This is a discovery question. If the polynomial is linear, solve the single equation. If the polynomial is quadratic and factorable, factorise it and set each factor equal to zero. If factorisation looks awkward but the question only asks for sum or product, do not force factorisation. Read the wording carefully. A question that says ‘find zeros’ expects the actual values of x; a question that says ‘find sum of zeros’ expects a relationship calculation.

When the question gives a graph

Do not try to rebuild the whole equation from the graph unless the question asks for it. First locate the x-axis. Then read the x-coordinates of the points where the graph meets that axis. Those x-values are the zeros. If the graph merely touches and turns back, count that as one repeated zero if the question discusses nature or number of zeros. If the graph never meets the x-axis, write that it has no real zero from the graph shown.

When the question asks for a polynomial from zeros

Place the zeros in the form alpha and beta. Find alpha + beta and alpha beta separately. Then write x^2 - (alpha + beta)x + alpha beta. After that, simplify signs and clear fractions if needed. If a leading coefficient is specified, multiply the monic polynomial by that value. If no leading coefficient is specified, the monic polynomial is usually the neatest answer.

When an unknown coefficient appears

Underline what is given: sum of zeros, product of zeros, or one of the zeros. If sum is given, set -b/a equal to that value. If product is given, set c/a equal to that value. If one zero is given, substitute it directly into the polynomial because a zero makes the polynomial value equal to zero. These three routes cover most unknown-coefficient questions in this chapter.

When coefficients are written in unusual order

Rewrite the polynomial mentally in descending powers before using formulas. For 7 - 4x + 2x^2, read it as 2x^2 - 4x + 7. Therefore a = 2, b = -4, and c = 7. The order of writing does not change the polynomial, but it can trick you into assigning the wrong coefficient if you rush.

When the constant term is zero

If a polynomial has a factor x, then 0 is one of its zeros. For example, x^2 - 5x = x(x - 5), so the zeros are 0 and 5. This also matches the product relationship because the constant term is zero, so the product of zeros is zero. At least one zero must then be zero. This is a quick recognition pattern.

When both zeros are equal

Repeated zeros usually come from a perfect square quadratic. If x^2 - 10x + 25 = (x - 5)^2, the zero is 5 twice. In coefficient language, the sum is 10 and the product is 25, which matches 5 + 5 and 5 x 5. If a graph is shown, this often appears as a parabola touching the x-axis at one point.

When a value is not a zero

A number either makes the polynomial zero or it does not. If p(1) equals -4, then 1 is not a zero. Do not adjust the value or continue solving unless the question asks for something else. A clear negative answer is still a complete answer when supported by substitution. In exams, write the non-zero result and the conclusion in words.

When the answer has many possible forms

Polynomial answers can look different but be equivalent. For zeros 2 and 3, x^2 - 5x + 6, 2x^2 - 10x + 12, and -x^2 + 5x - 6 all have the same zeros. If the question does not demand a particular leading coefficient, give the simplest form. If you are checking someone else’s answer, divide out any common non-zero multiplier before deciding it is different.

Full Revision Walkthrough: From First Reading to Final Check

This walkthrough is designed for the day before a class test. Read it once, then solve five mixed questions. The aim is to make your method choice automatic without turning the chapter into memorised steps.

Step 1: Translate the expression into parts

Before solving anything, read the polynomial like a sentence. In 6x^2 - 11x + 3, the quadratic term is 6x^2, the linear term is -11x, and the constant term is 3. This gives a = 6, b = -11, and c = 3. If you begin with this translation, every formula later becomes safer. If you skip it, the most common error is to treat b as 11 instead of -11.

Step 2: Decide whether the task is evaluation or solving

Substitution questions are evaluation tasks. They ask, in effect, what is p(a)? Zero-finding questions are solving tasks. They ask for all values of x that make p(x)=0. The two tasks use similar arithmetic but have different endpoints. If the question says, ‘Check whether 2 is a zero,’ do not solve the whole quadratic. Just evaluate p(2). If the question says, ‘Find the zeros,’ do not stop after one substitution. Solve the equation.

Step 3: Use factorisation only when it answers the question

Factorisation is powerful because of the zero product idea: if AB = 0, then A = 0 or B = 0. But it is not always the shortest path. If a question asks only for the sum of zeros of 7x^2 - 3x + 1, you can write -b/a = 3/7 immediately. Factorising first would be extra work and may even be difficult. Good exam technique means using the method that matches the wording.

Step 4: Connect factors to zeros slowly

A factor tells you how to get a zero, but the sign must be handled through an equation. From (x - 8), write x - 8 = 0, so x = 8. From (2x + 5), write 2x + 5 = 0, so x = -5/2. This one-line solving habit prevents the common mistake of changing signs by memory. It also handles factors like 3x - 7, where the zero is 7/3, not just 7.

Step 5: Check the number of zeros against the degree

A degree-two polynomial cannot have three different real zeros. A degree-one polynomial cannot have two different zeros. This rule is useful for checking. If your factorisation of a quadratic seems to produce three answers, something has gone wrong in algebra. If a graph of a cubic appears to cross the x-axis three times, that is possible; if a quadratic graph does so, it is not. The degree gives a ceiling, not a promise.

Step 6: Make the graph meaning practical

You do not need advanced graphing to understand zeros. The x-axis is the line where y is zero. The graph of y=p(x) reaches that axis exactly when p(x)=0. If the graph crosses at x=4, then p(4)=0. If it crosses at x=-2, then p(-2)=0. A graph question is often just a zero question drawn as a picture.

Step 7: Read coefficient relationships both ways

The relationship is useful in two directions. From a polynomial, it gives sum and product of zeros. From zeros, it helps you build a polynomial. For x^2 - 9x + 20, sum of zeros is 9 and product is 20. From zeros 4 and 5, sum is 9 and product is 20, so the polynomial is x^2 - 9x + 20. Seeing the symmetry makes the formula easier to remember.

Step 8: Treat fractions as normal numbers

Fractional zeros look harder because the arithmetic takes more space, but the logic is unchanged. If zeros are 2/3 and -5, the sum is 2/3 - 5 = -13/3 and the product is -10/3. A monic polynomial is x^2 + (13/3)x - 10/3. Multiplying by 3 gives 3x^2 + 13x - 10. The fraction has not changed the concept.

Step 9: Use signs for sense-checking

Signs tell a story. A positive product means the two zeros have the same sign, unless one is complex outside the real graph discussion. A negative product means one zero is positive and one is negative. A positive sum means the positive side is larger overall; a negative sum means the negative side is larger overall. These are not substitutes for exact solving, but they help you notice impossible answers quickly.

Step 10: Write answers in exam language

A neat answer has three parts: method, calculation, and conclusion. For example: ‘Given polynomial is 2x^2 - 9x + 10. Here a=2, b=-9, c=10. Therefore sum of zeros = -b/a = 9/2 and product of zeros = c/a = 5.’ This format is short, but it tells the examiner exactly what relationship you used and why your answer follows.

Step 11: Revise by mixing question types

Do not practise ten identical factorisation questions and assume the chapter is complete. Mix the tasks: one identification question, one substitution check, one factorisation, one coefficient relationship, one polynomial-from-zeros question, one graph-reading question, and one unknown-coefficient question. The exam can switch between these without warning. Mixed practice trains your method selection, which is the real skill of this chapter.

Step 12: Build a final two-minute checklist

Before submitting a solution, ask four quick questions. Did I carry the sign of b? Did I divide by a? If I found factors, did I convert them into zeros? If I constructed a polynomial, did I simplify it and clear fractions where useful? This checklist is short enough to use during a test and specific enough to catch most Class 10 polynomial mistakes.

Five-Minute Oral Drill Before a Test

This drill is for fast recall. It should not replace written practice, but it is excellent for the last revision slot before school because it strengthens vocabulary, signs, and formula memory without needing a full notebook session.

Say the role of each coefficient

For ax^2 + bx + c, say aloud: a controls the quadratic term and must not be zero, b controls the linear term, and c is the constant term. Then say the two relationships: sum of zeros is -b/a, product of zeros is c/a. This oral repetition is useful because the formulas are short but sign-sensitive. If you can say them correctly without looking, you are less likely to freeze during a test.

Convert factors into zeros

Ask someone to call out factors such as x - 9, x + 4, 2x - 3, and 5x + 1. Your job is to answer zeros quickly: 9, -4, 3/2, and -1/5. This drill trains the equation step in your head. It also helps with factorisation questions where students often write the factors correctly but then lose marks while extracting the zeros.

Build polynomials from pairs

Use small pairs first: 2, 3; -1, 6; -4, -5; 1/2, 7. For each pair, say the sum, the product, and the monic polynomial. For -1, 6, the sum is 5, product is -6, and the polynomial is x^2 - 5x - 6. This connects arithmetic with structure and makes the construction formula feel natural.

Check answers by sum and product

Take any proposed zeros and test them against the polynomial. If someone says 2 and 5 are zeros of x^2 - 7x + 10, check sum 7 and product 10. If someone says -2 and -5, the product still matches but the sum is -7, so the answer is wrong. This type of checking is quick and catches sign reversals immediately.

Read a graph sentence

Practise saying: ‘The graph meets the x-axis at these x-values, so these x-values are the zeros.’ This sentence prevents two common errors. First, it stops you from reading the y-intercept as a zero. Second, it reminds you that a graph question is still an algebra question about p(x)=0. If the graph touches the x-axis only once, say that the polynomial has one repeated zero at that x-value.

Explain one answer like a tutor

Choose one solved question and explain it in three sentences. Sentence one: identify the polynomial and the method. Sentence two: show the calculation. Sentence three: state the conclusion. For example, ‘Here a=3, b=-8, and c=4, so I will use the coefficient relationship. Sum of zeros is -b/a = 8/3 and product is c/a = 4/3. Therefore the zeros have sum 8/3 and product 4/3.’ If you can explain it, you understand it.

Teacher and Parent Note: How to Support This Chapter

If a student is weak in Polynomials, the most useful support is not giving more formulas at once. Start by asking the student to explain what a zero means in their own words. Then ask them to check one value by substitution. Once this is steady, move to simple factorisation and only then to the relationship between zeros and coefficients. The order matters because the relationship is easier to remember when the student already understands that zeros are inputs, not outputs.

For home revision, avoid turning every session into a long worksheet. Ten minutes of mixed questions is often better than forty minutes of one repeated pattern. A balanced mini-session can include one degree question, one substitution check, one factorisation question, one sum-product question, and one polynomial-from-zeros question. The student should say the method before solving. This reveals whether the problem is algebra skill or question reading.

When checking work, look for three habits: coefficients written with signs, answers concluded in words, and at least one verification line. These habits are small, but they make the chapter reliable. A student who writes a, b, and c correctly will usually recover from small arithmetic pressure. A student who skips that setup may know the formula but still lose marks because of one sign.

The strongest signal of readiness is transfer. After the student solves a familiar question, change one feature: make one zero negative, make the leading coefficient greater than one, or ask for the polynomial instead of the zeros. If the student can still choose the right method, the concept is stable and ready for a timed class test with mixed problems.

For students who are preparing independently, one useful weekly habit is an error notebook. Write the wrong answer, the correct answer, and the exact reason for the correction. In Polynomials, most reasons will be simple: wrong sign of b, missed division by a, factor not converted into a zero, or graph crossing read from the wrong axis. Seeing the same mistake twice is the signal to practise that one micro-skill.