Polynomials Class 10

Polynomials is Chapter 2 of NCERT Class 10 Mathematics. It focuses on the geometrical meaning of zeroes, relationships between zeroes and coefficients, construction of polynomials from zeroes and the polynomial division algorithm. Teachoo provides solutions for Exercises 2.1 and 2.2, examples, difficult questions, case-based practice, exemplar MCQs and past-year board MCQs.

Zeroes and their geometric meaning

A zero of polynomial p(x) is a value α for which p(α) = 0. On the graph y = p(x), real zeroes are the x-coordinates where the graph meets or touches the x-axis. A linear polynomial has at most one real zero, a quadratic at most two and a cubic at most three, though fewer real zeroes are possible.

The graph gives a visual interpretation, while substitution verifies a proposed zero algebraically. Students should distinguish a polynomial’s zero from its constant term and from the y-intercept.

Relationship between zeroes and coefficients

For quadratic polynomial ax² + bx + c with zeroes α and β:

• α + β = −b/a;

• αβ = c/a.

For cubic polynomial ax³ + bx² + cx + d with zeroes α, β and γ:

• α + β + γ = −b/a;

• αβ + βγ + γα = c/a;

• αβγ = −d/a.

These relationships can verify zeroes or construct a polynomial when roots are known. A monic quadratic with zeroes α and β is x² − (α + β)x + αβ; any non-zero constant multiple has the same zeroes.

Polynomial division algorithm

For polynomials p(x) and non-zero g(x), division gives p(x) = g(x)q(x) + r(x), where the degree of r is less than the degree of g. Students use this to divide polynomials, verify results and find unknown coefficients or zeroes in suitable questions.

Topics available on Teachoo

• Exercises 2.1 and 2.2 and NCERT examples;

• geometrical meaning of zeroes;

• relations between zeroes and coefficients;

• forming polynomials using zeroes;

• division algorithm;

• finding zeroes through division;

• difficult polynomial questions;

• case-based questions, exemplar MCQs and past-year MCQs.

Learning outcomes

Students should be able to identify real zeroes from graphs, verify them by substitution, apply coefficient relations and form a polynomial from specified zeroes. They should divide polynomials correctly and check dividend = divisor × quotient + remainder.

Why is this chapter important?

Polynomials underpin quadratic equations, graphs, factorisation and higher algebra. Board questions often combine two ideas—for example, a relationship among zeroes followed by formation of a new polynomial—so conceptual connections matter.

How Teachoo helps

Teachoo separates graphical interpretation, coefficient relations and division. First identify the polynomial’s degree and coefficients, including zero coefficients for missing terms. Write the relevant relation before substitution. For division, arrange both polynomials in descending powers and insert missing-power placeholders when needed.

After NCERT exercises, practise graphical, case-based and difficult mixed questions. Check every derived polynomial by comparing its required sum and product of zeroes.

Board-exam and competency preparation

Board questions may provide a graph, a relationship between zeroes, an unknown coefficient or a polynomial-division condition. First identify the degree and write coefficients in descending order, including zeros for missing powers. If a graph is given, count x-axis intersections carefully and distinguish crossing from touching—both indicate a real zero.

When new zeroes are expressed in terms of old ones, calculate their sum and product symbolically before forming the polynomial. In a case-based problem, do not rush from the story to a formula; state what the variable and polynomial represent. For division-algorithm questions, comparing coefficients after expansion is often cleaner than repeated substitution.

Quick revision checklist

Read zeroes from three different graphs, verify coefficient relations for a quadratic and cubic, form polynomials from transformed zeroes and complete one long polynomial division. Expand the final divisor–quotient–remainder identity to check every answer.

Common mistakes to avoid

Do not forget the negative sign in α + β = −b/a. A graph’s y-intercept is not generally a zero. When forming a polynomial, remember that non-zero multiples have the same roots. In division, the remainder’s degree must be smaller than the divisor’s degree.

Deeper reasoning and concept connections

The strongest way to learn Polynomials is to separate three layers: the object being studied, the rule that describes it and the reason the rule works. A correct numerical result is useful, but a complete mathematical answer also explains the relationship used. Students should compare examples and non-examples, change one condition at a time and observe whether the conclusion still holds.

This chapter is part of a longer progression. Its vocabulary and representations will appear again in algebra, geometry, data, measurement or higher problem-solving. Build links deliberately: translate pictures into statements, statements into operations and operations back into a sensible interpretation. If the final result cannot be explained in ordinary language, the method has probably been followed mechanically rather than understood.

How to solve unfamiliar and competency-based questions

When a question looks new, do not search memory for an identical example. Classify it. Decide whether it asks for recognition, calculation, representation, comparison, explanation or proof. Write the relevant definition or property first. Next, organise the data and select the shortest valid method. This converts an unfamiliar surface story into a familiar mathematical structure.

Use estimation and special cases as quality checks. Test zero, one, equal values, endpoints or a simple symmetric figure whenever they are permitted. A result that violates the diagram, scale, sign, unit or expected range is a signal to recheck the setup. In multi-part cases, carry forward only verified results so one early error does not silently contaminate every later answer.

What complete mastery looks like

For Polynomials, a student should be able to define the central ideas in simple language, recognise them in different representations, solve routine questions accurately and explain the method used. They should also be able to correct a flawed solution, create an example satisfying given conditions and combine two ideas from the chapter in one problem. A reliable mastery test is to solve one direct question, one application question and one reasoning question without looking at notes, then explain all three aloud or in writing.

Keep a compact error log with four labels: concept, interpretation, calculation and presentation. Reattempt each error after a gap instead of rereading the answer immediately. Improvement comes from correcting the decision that caused the mistake, not from repeating questions whose method is already known.

Additional frequently asked questions

What should a student know before starting Polynomials?

Revise the definitions, number operations, diagrams or notation used at the beginning of the chapter. The prerequisite list should be short: if an earlier skill blocks progress, repair that skill with two or three focused questions and return to the chapter.

How can a student check an answer in Polynomials?

Use an independent check whenever possible: substitute the result, reverse the operation, estimate its size, compare it with the figure, test a simpler case or solve using another representation. A check should examine the mathematical condition, not merely repeat the same arithmetic.

How many questions are enough for strong preparation?

There is no fixed number. Stop counting questions and track coverage: every concept, every standard method, at least one mixed problem, one competency-based problem and every previously incorrect type should be solved independently. Ten varied, analysed questions are more valuable than fifty copied solutions.

How should Teachoo solutions be used without becoming dependent on them?

Attempt the question first and mark the exact step where progress stops. Read only enough of the solution to repair that step, close it and restart the question. Finally, solve a similar problem without help. This turns a solution into feedback rather than a substitute for thinking.

Frequently asked questions

What is a zero of a polynomial?

It is a value of the variable that makes the polynomial equal to zero.

How are zeroes visible on a graph?

Real zeroes are the x-coordinates where y = p(x) intersects or touches the x-axis.

How can polynomial division be checked?

Verify that dividend equals divisor multiplied by quotient plus remainder.

Connect the graph, roots and coefficients. Treating them as one system makes polynomial questions far easier.