p(x)

Chapter 2 · Class X Mathematics · MCQ Practice

MCQ Practice Arena

Polynomials

Zeroes, Graphs and Coefficients — Master the Relationship That Scores

📋 50 MCQs ⭐ 28 PYQs ⏱ 55 sec/Q

🚀 Start Practising 📖 Revise Notes First

MCQ Bank Snapshot

50Total MCQs

20Easy

22Medium

8Hard

28PYQs

55 secAvg Time/Q

6Topics

Easy 40% Medium 44% Hard 16%

Why Practise These MCQs?

CBSE Class XNTSEState Boards

Polynomials MCQs are among the most direct in Class X — the relationship between zeroes and coefficients appears in both 2-mark and MCQ formats every CBSE year. NTSE algebraic reasoning uses polynomial factorisation. Graph-based zeroes questions (count x-intercepts) are fast, visual marks. Division algorithm MCQs are a CBSE favourite for 3-mark questions.

Topic-wise MCQ Breakdown

Zeroes of a Polynomial (Concept)6 Q

Graphical Meaning of Zeroes8 Q

Sum & Product of Zeroes (Quadratic)14 Q

Sum/Product/Symmetric — Cubic7 Q

Division Algorithm for Polynomials8 Q

Form Polynomial Given Zeroes7 Q

Must-Know Formulae Before You Start

Recall these cold before attempting MCQs — they appear in >70% of questions.

$\text{Quadratic: } \alpha+\beta = -b/a,\ \alpha\beta = c/a$

$\text{Cubic: } \alpha+\beta+\gamma = -b/a$

$\alpha\beta+\beta\gamma+\gamma\alpha = c/a,\ \alpha\beta\gamma = -d/a$

$p(x) = g(x)\cdot q(x) + r(x)\ (\text{Division Algorithm})$

$\text{Quadratic from zeroes: } x^2-(\alpha+\beta)x+\alpha\beta$

MCQ Solving Strategy

For sum/product MCQs, read the polynomial carefully — make sure it is in standard form ax²+bx+c before applying the formulae. For graph-based zeroes, count how many times the curve crosses the x-axis. For division algorithm MCQs, verify: p(x) = g(x)·q(x) + r(x) and degree of r(x) < degree of g(x). "Form a polynomial" MCQs: write x² − (sum)x + product directly.

⚠ Common Traps & Errors

⚠Zeroes of a polynomial ≠ coefficients — evaluate p(x) = 0
⚠Cubic: αβγ = −d/a, NOT +d/a
⚠If α is a zero, (x − α) is a factor — not (x + α)
⚠Degree of remainder must be strictly less than divisor degree
⚠Polynomial with zeroes 2, 3: x²−5x+6, not x²+5x+6

Difficulty Ladder

Work through each rung in order — do not jump to Hard before mastering Easy.

① Easy

Identify zeroes from graph, verify given zeroes by substitution

② Medium

Apply sum/product formulae, find k given a zero

③ Hard

Cubic zero relations, division algorithm with unknown coefficients

★ PYQ

CBSE — find α²+β², form polynomial; NTSE — factorisation reasoning

Continue Your Preparation

📖 Chapter Notes Revise concepts & theory 🧠 MCQ Practice Topic-wise Questions 📘✔️ Exercises Step-by-step NCERT solutions ✔️❌ True / False Concept-based True & False ➡ Next Chapter Pair of Linear Equations in Two Variables

🎯 Knowledge Check

Maths — POLYNOMIALS

50 Questions Class 10 MCQs

A polynomial of degree 0 is called a ____

a a. Linear polynomial b b. Constant polynomial c c. Quadratic polynomial d d. Zero polynomial

The degree of the polynomial $5x^4 - 3x^2 + 2x - 7$ is ____

a a. 4 b b. 3 c c. 2 d d. 1

The degree of the zero polynomial is ____

a a. 0 b b. 1 c c. Not defined d d. Infinite

The coefficient of $x^2$ in $5x^3 + 7x^2 - 4x + 9$ is ____

a a. 9 b b. -4 c c. 7 d d. 5

A linear polynomial has degree ____

a a. 0 b b. 1 c c. 2 d d. 3

Which of the following is a quadratic polynomial?

a a. $3x + 5$ b b. $x^2 + 2x + 1$ c c. $x^3 + 3x^2$ d d. $2x^4 - 5$

The zero of the polynomial $p(x) = x - 3$ is ____

a a. 0 b b. 1 c c. 3 d d. -3

The zero of $p(x) = 2x + 5$ is ____

a a. -5 b b. 5 c c. -5/2 d d. 5/2

The number of zeros of a cubic polynomial is ____

a a. 1 b b. 2 c c. 3 d d. 4

The zeros of the polynomial $p(x) = x^2 - 1$ are ____

a a. 0, 1 b b. -1, 1 c c. 1, 2 d d. -1, 0

The value of $p(x) = x^2 - 2x + 3$ at $x = 2$ is ____

a a. 3 b b. 5 c c. 7 d d. 11

When $p(x)$ is divided by $x - a$, the remainder is $p(a)$. This is called ____

a a. Factor theorem b b. Remainder theorem c c. Division algorithm d d. Euclid theorem

If $p(x) = x^2 + 2x + 1$, zeros are ____

a a. 1, -1 b b. 0, 1 c c. -1, -1 d d. 1, 1

The sum of zeros of $x^2 - 5x + 6$ is ____

a a. 5 b b. -5 c c. -6 d d. 6

The product of zeros of $2x^2 + 5x + 3$ is ____

a a. 3 b b. 5/2 c c. 3/2 d d. -3

If zeros of $x^2 - 7x + 10$ are $\alpha$ and $\beta$, then $\alpha + \beta =$ ____

a a. 10 b b. 7 c c. -7 d d. -10

The polynomial whose zeros are 2 and 3 is ____

a a. $x^2 - 5x + 6$ b b. $x^2 - 6x + 5$ c c. $x^2 + 5x + 6$ d d. $x^2 - x + 1$

If zeros are equal, the discriminant $b^2 - 4ac$ is ____

a a. > 0 b b. = 0 c c. < 0 d d. Any value

If $x = 1$ is a zero of $x^3 - 3x^2 + x + 1$, then the remainder is ____

a a. 0 b b. 1 c c. 2 d d. -1

Factorise $x^2 - 16$.

a a. $(x - 4)^2$ b b. $(x - 4)(x + 4)$ c c. $(x + 8)(x - 2)$ d d. None

Polynomial having zeros at -2 and 5 is ____

a a. $x^2 - 3x - 10$ b b. $x^2 + 3x - 10$ c c. $x^2 - 10x + 25$ d d. $x^2 - 7x + 10$

Zeros of $x^2 + 9$ are ____

a a. 3, -3 b b. 9, -9 c c. 3i, -3i d d. None

If one zero of $2x^2 + 3x - 5$ is 1, the other is ____

a a. 5/2 b b. -5/2 c c. 1 d d. -4

A cubic polynomial has maximum how many zeros?

a a. 2 b b. 3 c c. 4 d d. 1

The remainder when $x^3 - 2x^2 + 4x - 8$ is divided by $x - 2$ is ____

a a. 0 b b. 8 c c. -8 d d. 4

$p(x)$ is divisible by $x - 3$ if ____

a a. $p(3) = 0$ b b. $p(0) = 0$ c c. $p(-3) = 0$ d d. $p(1) = 0$

Zeros of $x^2 - 3x + 2$ are ____

a a. 1, 2 b b. -1, -2 c c. 2, 3 d d. 0, 2

For $p(x) = x^2 + 4x + 3$, sum of zeros = ____

a a. -4 b b. 4 c c. -3 d d. 3

For $p(x) = x^2 - 2x + 1$, zeros are ____

a a. 0, 1 b b. 1, 1 c c. -1, -1 d d. 2, 2

If a polynomial is divisible by both $x - 1$ and $x + 2$, then $p(1)$ and $p(-2)$ are ____

a a. Both 0 b b. Both 1 c c. Both 2 d d. None

For $x^2 - 2kx + (k^2 - 1) = 0$, equal roots are possible when ____

a a. k = 1 b b. k = -1 c c. Any k d d. None

If $p(x)$ divided by $x - 1$ gives remainder 5, find $p(1)$.

a a. 0 b b. 5 c c. -5 d d. 1

If zeros of $x^2 + 7x + 10$ are a, ß, find aß.

a a. 7 b b. 10 c c. -10 d d. -7

The number of terms in $3x^2 + 2x - 7$ is ____

a a. 2 b b. 3 c c. 4 d d. 1

The degree of $5y^3 - 4y^5 + 3y$ is ____

a a. 3 b b. 4 c c. 5 d d. 2

If $p(x) = x^2 - 4$, then zeros are ____

a a. -2, 2 b b. 0, 4 c c. -4, 4 d d. 2, 4

For $p(x) = ax^2 + bx + c$, if one zero = 0, then $c$ = ____

a a. 0 b b. a c c. b d d. 1

The graph of a quadratic polynomial is a ____

a a. Line b b. Circle c c. Parabola d d. Ellipse

The graph of a linear polynomial is a ____

a a. Straight line b b. Circle c c. Parabola d d. Hyperbola

If the graph of polynomial $p(x)$ touches x-axis at one point, it has ____

a a. 1 zero b b. 2 zeros c c. 3 zeros d d. None

The product of zeros of $3x^2 - 2x - 1$ is ____

a a. 1/3 b b. -1/3 c c. -1 d d. 3

If zeros of $x^2 - kx + 9$ are equal, then $k $= ____

a a. 0 b b. 3 c c. 6 d d. 9

If $x = 2$ is zero of $x^3 - 2x^2 + 4x - 8$, quotient polynomial is ____

a a. $x^2 + 4$ b b. $x^2 + 4x + 2$ c c. $x^2 + 4x$ d d. $x^2 + 4$

For $p(x) = 3x^2 - 5x + 2$, sum of zeros = ____

a a. 5 b b. -5 c c. 5/3 d d. 5/2

A polynomial of degree 3 is called ____

a a. Quadratic b b. Linear c c. Cubic d d. Biquadratic

If zeros are 1 and -3, the polynomial is ____

a a. $x^2 + 2x - 3$ b b. $x^2 - 2x - 3$ c c. $x^2 + 3x - 2$ d d. $x^2 - x + 3$

$x^2 - 4x + 3 = 0$ has how many zeros?

a a. 0 b b. 1 c c. 2 d d. Infinite

Zeros of $x^3 - 6x^2 + 11x - 6$ are ____

a a. 1, 2, 3 b b. -1, -2, -3 c c. 1, 3, 6 d d. None

For quadratic $ax^2 + bx + c$, the relationship between coefficients and zeros is ____

a a. Sum = $\frac{b}{a}$, Product = $\frac{c}{a}$ b b. Sum = $-\frac{b}{a}$, Product = $\frac{c}{a}$ c c. Sum = $\frac{-a}{b}$, Product =$\frac{c}{a}$ d d. None

If zeros of polynomial are 4 and 5, find polynomial with leading coefficient 2.

a a. $2x^2 - 18x + 40$ b b. $2x^2 - 9x + 10$ c c. $2x^2-18x + 20$ d d. $x^2-9x+20$

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Frequently Asked Questions

A polynomial is an algebraic expression that combines variables and numbers, using only non-negative whole number exponents.

Polynomials are classified by their highest exponent: linear (power one), quadratic (power two), cubic (power three), and higher-degree polynomials.

The degree of a polynomial is the largest exponent of the variable found in the polynomial.

A linear polynomial is an expression with the variable raised to one, for example, "a times x plus b."

A quadratic polynomial includes the variable raised to the second power, like "a times x squared plus b times x plus c."

A cubic polynomial contains the variable raised to the third power, such as "a times x cubed plus b times x squared plus c times x plus d."

The coefficient is the number multiplied by the variable in each term, for example, in "four x squared," the number four is the coefficient.

You add polynomials by merging terms that have the same variables and powers, using regular addition for their coefficients.

Subtracting polynomials means you subtract the coefficients of terms that have matching variables and exponents.

To multiply polynomials, multiply every term in one polynomial by every term in the other and then add any like terms.

The zero of a polynomial is a value for the variable that makes the whole expression equal to zero.

The Factor Theorem says if a polynomial equals zero when you substitute a number for the variable, then the expression "variable minus that number" is a factor of the polynomial.

The Remainder Theorem tells us that if you divide a polynomial by "variable minus a number," the remainder is what you get when you plug that number into the polynomial.

To factorize a polynomial, rewrite it as a multiplication of simpler polynomials, just like splitting a number into its factors.

Polynomials are crucial because they help in describing patterns, solving equations, and modeling real-life scenarios in mathematics and science.

POLYNOMIALS – Learning Resources

Detailed Notes

True / False

NCERT Solutions

ACADEMIA AETERNUM तमसो मा ज्योतिर्गमय · Est. 2025

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Polynomials

MCQ Bank Snapshot

Why Practise These MCQs?

Topic-wise MCQ Breakdown

Must-Know Formulae Before You Start

MCQ Solving Strategy

⚠ Common Traps & Errors

Difficulty Ladder

Continue Your Preparation

Maths — POLYNOMIALS

Frequently Asked Questions

Q 01 What is a polynomial?

Q 02 What are the types of polynomials?

Q 03 What is the degree of a polynomial?

Q 04 What is a linear polynomial?

Q 05 What is a quadratic polynomial?

Q 06 What is a cubic polynomial?

Q 07 What is the coefficient in a polynomial?

Q 08 How do you add polynomials?

Q 09 How do you subtract polynomials?

Q 10 How do you multiply polynomials?

Q 11 What is the zero of a polynomial?

Q 12 What is the Factor Theorem?

Q 13 What is the Remainder Theorem?

Q 14 How can polynomials be factorized?

Q 15 What is the importance of polynomials in mathematics?

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POLYNOMIALS – Learning Resources

Detailed Notes

True / False

NCERT Solutions

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