How many practice questions are available for Polynomials?

SparkEd Math offers 60+ practice questions for Polynomials across 3 difficulty levels (Easy, Medium, Hard) plus a timed test mode. Each question comes with visual step-by-step solutions and AI coaching hints.

Is Polynomials content free on SparkEd Math?

Yes! All Polynomials content on SparkEd Math is 100% free, including practice questions, visual solutions, lessons, and AI coaching. No credit card required.

How does SparkEd Math teach Polynomials differently from textbooks?

SparkEd Math uses visual step-by-step solutions for every Polynomials problem, making complex concepts easy to understand. The platform includes 3 difficulty levels for progressive learning, a Super Power Help AI coach for instant doubt-clearing, and interactive practice rather than passive reading.

Is SparkEd's Polynomials content aligned with the Olympiad syllabus?

Yes, all Polynomials questions and lessons on SparkEd Math are fully aligned with the Mathematical Olympiad Preparation (Olympiad) syllabus for Class 10. The content follows the official curriculum structure and exam pattern.

Polynomials — Class 10 Olympiad (2025-26)

About Polynomials — Class 10 Olympiad

Find zeros and verify relationships between zeros and coefficients; divide polynomials and solve competition problems. This topic is part of the Olympiad Class 10 mathematics syllabus (chapter: Module 2). On this page you can practice 50 questions across three difficulty levels — 20 easy, 10 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 28-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Polynomials — solved examples for Class 10 Olympiad

Example 1easy

If P(x) = (k²-4)x³ + (k+2)x² + 5x + 7 is a quadratic polynomial, what is the value of k?

A)-2
B)0
C)2
D)4

Step-by-step solution

For P(x) to be a quadratic polynomial, the coefficient of x³ must be zero, and the coefficient of x² must be non-zero.
Set the coefficient of x³ to zero: k² - 4 = 0. This implies k² = 4, so k = 2 or k = -2.
Now, check the coefficient of x²: (k+2). If k = -2, then k+2 = -2+2 = 0, which would make the polynomial linear, not quadratic. If k = 2, then k+2 = 2+2 = 4, which is non-zero.
Therefore, for P(x) to be a quadratic polynomial, the value of k must be 2.

Answer: 2

Example 2medium

If the roots of the polynomial x³ - 12x² + 44x - k = 0 are in arithmetic progression, find the value of k.

A)48
B)44
C)40
D)52

Step-by-step solution

Let the roots be (a-d), a, (a+d).
Sum of roots: (a-d) + a + (a+d) = -(-12)/1 = 12. This simplifies to 3a = 12, so a = 4.
Sum of products of roots taken two at a time: (a-d)a + a(a+d) + (a-d)(a+d) = 44/1 = 44. Substituting a=4: 4(4-d) + 4(4+d) + (4-d)(4+d) = 44. This simplifies to 16 - 4d + 16 + 4d + 16 - d² = 44, so 48 - d² = 44, which means d² = 4.
Product of roots: (a-d) × a × (a+d) = -(-k)/1 = k. Substituting a=4 and d²=4: k = 4(4² - d²) = 4(16 - 4) = 4(12) = 48.

Answer: 48

Example 3hard

If α and β are the roots of the quadratic polynomial P(x) = x² - (k+3)x + (3k-1), such that α² + β² = 20, what is the value of k?

Step-by-step solution

From Vieta's formulas for P(x) = x² - (k+3)x + (3k-1), the sum of roots is:
$α+β = k+3$
And the product of roots is:
$αβ = 3k-1$
We are given α² + β² = 20. Using the identity α² + β² = (α+β)² - 2αβ, we substitute the expressions:
$(k+3)² - 2(3k-1) = 20$
Expand and simplify the equation: k² + 6k + 9 - 6k + 2 = 20. This simplifies to k² + 11 = 20.
$k² + 11 = 20$
Solving for k: k² = 9, which gives k = ±3. Among the given options, 3 is present.
$k=3$

Answer: 3

Practice questions on Polynomials

Q1.easy
Let P(x) = x² - 3x + 2. What is the value of P(P(1))?
1. A)0
2. B)1
3. C)2
4. D)3
Show answer
Answer: 2
Hint: Start by evaluating the innermost expression, P(1), and then substitute that result back into P(x).
Q2.easy
If α and β are the zeros of the polynomial 2x² - 3x + 5, what is the value of (α+1)(β+1)?
1. A)3
2. B)4
3. C)5
4. D)6
Show answer
Answer: 5
Hint: Recall Vieta's formulas for the sum and product of zeros of a quadratic polynomial. Then, expand the given expression.
Q3.easy
If α and β are the zeros of the quadratic polynomial f(x) = x² + 2x - 15, find a quadratic polynomial whose zeros are 2α and 2β.
1. A)x² - 4x - 30
2. B)x² + 4x - 30
3. C)x² + 4x - 60
4. D)x² - 4x - 60
Show answer
Answer: x² + 4x - 60
Hint: First, find the sum and product of the original zeros. Then, use these to find the sum and product of the new zeros (2α and 2β).
Q4.medium
For the quadratic polynomial P(x) = ax² + bx + c (where a, b, c are real numbers, a ≠ 0), if one zero is the reciprocal of the other, which of the following statements must be true?
1. A)a = b
2. B)a = c
3. C)b = c
4. D)a + b + c = 0
Show answer
Answer: a = c
Hint: Let the zeros be α and 1/α. Use Vieta's formula for the product of zeros.
Q5.medium
A polynomial P(x) leaves a remainder of 5 when divided by (x - 2) and a remainder of 10 when divided by (x - 3). What is the remainder when P(x) is divided by (x - 2)(x - 3)?
1. A)5x + 5
2. B)5x - 10
3. C)5x - 5
4. D)5x
Show answer
Answer: 5x - 5
Hint: The remainder when dividing by a quadratic polynomial is at most a linear polynomial (Ax + B). Use the Remainder Theorem for P(2) and P(3).
Q6.medium
If the polynomials P(x) = x³ + ax² + bx + c and Q(x) = x³ + Ax² + Bx + C have a common root α, and if α is also a root of P(x) - Q(x), then which of the following must be true?
1. A)(a-A)α + (b-B) = 0
2. B)P(α)=Q(α)=0
3. C)a=A, b=B, c=C
4. D)(a-A)α² + (b-B)α + (c-C) = 0
Show answer
Answer: (a-A)α² + (b-B)α + (c-C) = 0
Hint: If α is a common root, then P(α)=0 and Q(α)=0. Consider the polynomial P(x) - Q(x).
Q7.hard
A polynomial P(x) leaves a remainder of 3 when divided by (x-1) and a remainder of 5 when divided by (x+1). What is the remainder when P(x) is divided by (x²-1)?
1. A)x+4
2. B)-x+4
3. C)2x+1
4. D)x-2
Show answer
Answer: -x+4
Hint: Use the Remainder Theorem to find P(1) and P(-1). Remember that when dividing by a quadratic, the remainder is at most a linear polynomial.
Q8.hard
Let the roots of the cubic polynomial P(x) = x³ - 6x² + kx - 6 be in an arithmetic progression. Find the value of k.
1. A)8
2. B)9
3. C)10
4. D)11
Show answer
Answer: 11
Hint: If the roots are in an arithmetic progression (AP), denote them as a-d, a, a+d. Use Vieta's formulas to relate the sum of roots and the product of roots to the coefficients.
Q9.hard
If the identity (x-1)(x-2)(x-3) = Ax(x-1)(x-2) + Bx(x-1) + Cx + D holds for all real values of x, find the value of A+B+C+D.
1. A)-2
2. B)-1
3. C)0
4. D)1
Show answer
Answer: -2
Hint: This is a polynomial identity. Evaluate both sides of the identity at specific, strategically chosen values of x (e.g., x=0, 1, 2, 3) to easily determine the coefficients.

These are 9 of the 50 questions available for Polynomials. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.