Class 10 Maths · Original EZT practice
Polynomials (Class 10 Maths Chapter 2): zeroes, graphs and two small formulas that do big work
A polynomial's zeroes are the x-values where it equals zero - the exact points where its graph meets the x-axis. This chapter teaches you to find zeroes, read them off a graph, and check them with two short relations between zeroes and coefficients. Below: the ideas in plain words, then original EZT practice with worked answers. (Note: this is original practice, not a textbook exercise key.)
10 original worked examples, 10 fresh self-check questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 17 October 2026:
1. Polynomials and degree
A polynomial in x is an expression like 3x^2 - x + 5 built from powers of x with whole-number exponents. The highest power is the degree: degree 1 is linear (3x + 2), degree 2 is quadratic (x^2 - 5x + 6), degree 3 is cubic. Constants like 7 are degree 0.
2. Zeroes: definition and finding them
A zero of p(x) is a value a with p(a) = 0. For a quadratic, factor or use the sum-product check below.
3. Reading zeroes from a graph
The graph of a quadratic p(x) is a parabola. Each point where it crosses or touches the x-axis is a zero: two crossings = two distinct zeroes; touching at one point = one repeated zero; floating above or below the axis = no real zeroes.
4. Zeroes and coefficients: the two relations
For ax^2 + bx + c = 0 with zeroes alpha and beta: alpha + beta = -b/a and alpha × beta = c/a. Use them to check answers, find a missing zero, or build a polynomial from given zeroes.
5. Cubics in one line (extension)
For ax^3 + bx^2 + cx + d with zeroes p, q, r: p + q + r = -b/a; pq + pr + qr = c/a; pqr = -d/a. Example check: x^3 - 6x^2 + 11x - 6 has zeroes 1, 2, 3 - sum 6, pairwise sum 2 + 3 + 6 = 11, product 6.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Name a polynomial by its degree
Choose the method: Find the highest whole-number power of x. Degree 1 is linear, degree 2 is quadratic, and degree 3 is cubic.
EZT practice question
Name the degree of (a) 5x - 7, (b) 2x^2 + x - 9, (c) x^3 - 6x^2 + 11x - 6.
Show the worked solution
- The highest powers are x, x^2 and x^3 respectively.
- Match each degree to its name.
Answer: (a) 1 - linear; (b) 2 - quadratic; (c) 3 - cubic.
Check: Only the highest exponent decides the degree; lower terms do not change it.
Watch for this: Do not count the number of terms as the degree.
Now try it yourself
Name the degree of 7x^4 - 2x + 1.
Check your answer
4
The highest power is x^4.
Check whether an expression is a polynomial
Choose the method: Every power of the variable must be a whole number. Negative or fractional powers are not allowed.
EZT practice question
Is x + 1/x a polynomial?
Show the worked solution
- Rewrite 1/x as x^(-1).
- A negative exponent is not a whole-number exponent.
Answer: No.
Check: Polynomials may contain constants and non-negative whole-number powers only.
Watch for this: A term can look algebraic and still fail the polynomial test.
Now try it yourself
Is x^2 + √x a polynomial?
Check your answer
No.
√x is x^(1/2), and 1/2 is not a whole-number exponent.
Find zeroes by factorising
Choose the method: Factor the quadratic, set each factor to zero, then substitute each answer into p(x).
EZT practice question
Find the zeroes of p(x) = x^2 - 5x + 6.
Show the worked solution
- x^2 - 5x + 6 = (x - 2)(x - 3).
- So x - 2 = 0 or x - 3 = 0.
Answer: 2 and 3.
Check: p(2) = 4 - 10 + 6 = 0; p(3) = 9 - 15 + 6 = 0.
Watch for this: The factor x - 2 gives the zero x = 2, not x = -2.
Now try it yourself
Find the zeroes of x^2 - 7x + 12.
Check your answer
3 and 4.
The polynomial factors as (x - 3)(x - 4).
Factor when the leading coefficient is not 1
Choose the method: Split the middle term so the two products match ac, then factor by grouping.
EZT practice question
Find the zeroes of p(x) = 3x^2 - x - 4.
Show the worked solution
- 3x^2 - x - 4 = 3x^2 + 3x - 4x - 4.
- Group: 3x(x + 1) - 4(x + 1) = (3x - 4)(x + 1).
- So 3x - 4 = 0 or x + 1 = 0.
Answer: 4/3 and -1.
Check: 3(4/3)^2 - (4/3) - 4 = 16/3 - 4/3 - 12/3 = 0; 3(-1)^2 - (-1) - 4 = 0.
Watch for this: When 3x - 4 = 0, the zero is 4/3, not -4/3.
Now try it yourself
Find the zeroes of 2x^2 + 5x - 3.
Check your answer
1/2 and -3.
It factors as (2x - 1)(x + 3).
Use a given zero to find a parameter
Choose the method: If a is a zero, substitute x = a and solve p(a) = 0.
EZT practice question
For what k is -2 a zero of p(x) = x^2 + 3x + k?
Show the worked solution
- p(-2) = 4 - 6 + k.
- Set 4 - 6 + k = 0, so k = 2.
Answer: k = 2.
Check: Then p(x) = x^2 + 3x + 2 = (x + 1)(x + 2), whose zeroes are -1 and -2.
Watch for this: Substitute the x-value before solving; do not set k itself equal to the given zero.
Now try it yourself
For what k is 3 a zero of p(x) = x^2 - 5x + k?
Check your answer
k = 6.
p(3) = 9 - 15 + k = 0, so k = 6.
Build a quadratic from graph crossings
Choose the method: If the graph crosses the x-axis at r and s, one quadratic with those zeroes is (x - r)(x - s).
EZT practice question
A parabola crosses the x-axis at x = -2 and x = 4. Write a quadratic with these zeroes.
Show the worked solution
- Use factors (x + 2)(x - 4).
- Expand: x^2 - 2x - 8.
Answer: x^2 - 2x - 8.
Check: At x = -2, -2 + 2 = 0; at x = 4, 4 - 4 = 0.
Watch for this: A zero r gives the factor x - r, so a negative zero gives an x + factor.
Now try it yourself
A parabola crosses the axis at -5 and 1. Write the quadratic.
Check your answer
x^2 + 4x - 5.
(x + 5)(x - 1) = x^2 + 4x - 5.
Read a repeated zero from a graph
Choose the method: A parabola that touches the x-axis without crossing has one repeated zero.
EZT practice question
How many real zeroes does y = (x - 2)^2 have, and how does its graph look?
Show the worked solution
- (x - 2)^2 = 0 when x = 2.
- The same zero occurs twice, so the parabola touches the axis at 2.
Answer: One repeated zero, x = 2; the graph touches the x-axis without crossing.
Check: The square is positive on both sides of x = 2 and equals zero only there.
Watch for this: A repeated zero is not two distinct x-intercepts.
Now try it yourself
Describe the x-axis behaviour of y = (x + 3)^2.
Check your answer
It touches at x = -3 without crossing.
The zero -3 is repeated.
Check zeroes with sum and product
Choose the method: For ax^2 + bx + c, compare the zeroes with -b/a and c/a.
EZT practice question
Check the zeroes 4/3 and -1 from the previous factorisation against the relations.
Show the worked solution
- Sum = 4/3 - 1 = 1/3 = -(-1)/3.
- Product = (4/3)(-1) = -4/3 = c/a.
Answer: Both relations match.
Check: For 3x^2 - x - 4, a = 3, b = -1 and c = -4.
Watch for this: Use the sign of b carefully: the sum is -b/a, not b/a.
Now try it yourself
Check the zeroes 1/2 and -3 for 2x^2 + 5x - 3.
Check your answer
Sum -5/2 and product -3/2, both matching.
1/2 - 3 = -5/2 and (1/2)(-3) = -3/2.
Find a missing zero
Choose the method: Use the known zero to find the parameter, then use the sum of zeroes to find the other one.
EZT practice question
One zero of p(x) = x^2 - 4x + k is 1. Find k and the other zero.
Show the worked solution
- p(1) = 1 - 4 + k = 0, so k = 3.
- The sum of zeroes is -(-4)/1 = 4.
- Other zero = 4 - 1 = 3.
Answer: k = 3 and the other zero is 3.
Check: x^2 - 4x + 3 = (x - 1)(x - 3).
Watch for this: Do not stop after finding k; the question also asks for the other zero.
Now try it yourself
One zero of x^2 - 6x + k is 2. Find k and the other zero.
Check your answer
k = 8 and the other zero is 4.
p(2) = 4 - 12 + k = 0 gives k = 8; the zeroes sum to 6.
Build a polynomial from two zeroes
Choose the method: Use sum S and product P in x^2 - Sx + P. Clear fractions if integer coefficients are wanted.
EZT practice question
Form a quadratic whose zeroes are -3 and 1/2.
Show the worked solution
- Sum = -3 + 1/2 = -5/2; product = -3/2.
- x^2 - (-5/2)x - 3/2 = 0.
- Multiply by 2: 2x^2 + 5x - 3 = 0.
Answer: 2x^2 + 5x - 3 = 0.
Check: (x + 3)(2x - 1) = 2x^2 + 5x - 3.
Watch for this: The coefficient form is x^2 - Sx + P, so a negative sum becomes a positive x coefficient.
Now try it yourself
Form a quadratic with zeroes 2 and -1/4.
Check your answer
4x^2 - 7x - 2 = 0.
The sum is 7/4 and the product is -1/2; multiplying x^2 - (7/4)x - 1/2 by 4 clears fractions.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Find the zeroes of x^2 - 2x - 15.
- A parabola crosses the axis at -5 and 1. Write the quadratic.
- If the sum of the zeroes of 2x^2 - kx + 6 is 3, find k.
- How many zeroes does a graph crossing the axis three times show?
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Questions about this practice page
- What is a zero of a polynomial?
- A value a where p(a) = 0 - graphically, an x-coordinate where the curve meets the x-axis.
- How are zeroes related to the coefficients of a quadratic?
- Sum of zeroes = -b/a and product = c/a for ax^2 + bx + c = 0 - two relations that let you check or rebuild the polynomial.
- How do I find the zeroes of a quadratic?
- Factor it, or use the quadratic formula from Chapter 4, then confirm with the sum-product relations.
- How many zeroes can a degree-n polynomial have?
- At most n real zeroes - a quadratic at most 2, a cubic at most 3 - as many as the number of times its graph can meet the x-axis.
- Are these the NCERT Polynomials exercise questions?
- No. EZT pages are original practice on this chapter - use them to prepare for the textbook exercises, not as a substitute for your book.