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Polynomials Quiz - CBSE Class 10 Mathematics

Free practice quiz: Polynomials (CBSE Class 10 Mathematics). 13 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 10 Mathematics practice quiz has 13 multiple-choice questions on Polynomials, and 13 of them come with an explanation. Try the 5 sample questions below (tap an option to check yourself), then play the full quiz online. No sign-up needed.

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About this topic

A polynomial in x has only whole-number powers of x. Its degree tells the highest power and the maximum number of zeroes. A zero of p(x) is a value k with p(k) = 0, and it is a point where the graph of y = p(x) meets the x-axis.

Polynomial form: a_nxⁿ + … + a₁x + a₀, with n a whole number and a_n ≠ 0.
Degree = highest power of the variable. Linear = 1, quadratic = 2, cubic = 3.
Coefficient is the number multiplying a term. In 7x³ + 4x² − x + 5, the coefficient of x is −1.
Zero of p(x) is a number k such that p(k) = 0. Find it by substituting.
A polynomial of degree n has at most n zeroes.
Real zeroes of p(x) = x-coordinates of the points where the graph of y = p(x) meets (cuts or touches) the x-axis.
Linear graph is a straight line. Quadratic graph is a parabola, opening up if a > 0 and down if a < 0.

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Degree, Terms and Coefficients

A polynomial is a sum of terms like ax^n, where n is a whole number.
The degree is the highest power of the variable whose coefficient is not zero.
A coefficient is the number multiplying a power of x, and it includes the sign.

General polynomial of degree n

\[p(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_1x+a_0,\ a_n\neq 0\]

Quadratic polynomial

\[ax^2+bx+c,\ a\neq 0\]

Cubic polynomial

\[ax^3+bx^2+cx+d,\ a\neq 0\]

Common mistake: Counting the number of terms as the degree, or dropping the minus sign. In ax^n - bx, the coefficient of x is -b, not b.

Value and Zeroes of Polynomials

To find p(k), replace every x by k and simplify carefully.
k is a zero (root) of p(x) when p(k) = 0. A degree n polynomial has at most n zeroes.
If (x - a)^k is a factor, then a is a zero repeated k times. This is called multiplicity k.

Condition for a zero

\[p(k)=0\]

Difference of squares zeroes

\[x^2-a^2=0 \Rightarrow x=a \text{ or } x=-a\]

Zero a with multiplicity k

\[(x-a)^k\]

Sum and product of zeroes of ax^2+bx+c

\[\alpha+\beta=-\frac{b}{a},\quad \alpha\beta=\frac{c}{a}\]

Common mistake: Writing only the positive root of x^2 - a^2 and forgetting -a. Also, the zero of (x + a) is -a, not a.

Division, Remainder and Factor Theorem

Dividing p(x) by g(x) gives a quotient q(x) and a remainder r(x) of smaller degree.
When p(x) is divided by x - a, the remainder is simply p(a).
If p(a) = 0, then x - a is a factor of p(x), and a is a zero.

Division algorithm

\[p(x)=g(x)\,q(x)+r(x)\]

Remainder theorem

\[\text{Remainder of } p(x)\div(x-a)=p(a)\]

Factor theorem

\[p(a)=0 \iff (x-a) \text{ is a factor of } p(x)\]

Common mistake: Substituting the wrong sign. For division by x - a, substitute x = a. For division by x + a, substitute x = -a.

Identities, Factorisation and Real Zeroes

An identity is true for every value of the variables, not just for some values.
Expanding (a + b)^2 gives three terms. A perfect square trinomial factorises into two equal factors.
For ax^2 + bx + c, the discriminant tells whether the zeroes are real and whether they are equal.

Square of a sum

\[(a+b)^2=a^2+2ab+b^2\]

Square of a difference

\[(a-b)^2=a^2-2ab+b^2\]

Difference of squares

\[a^2-b^2=(a+b)(a-b)\]

Discriminant

\[D=b^2-4ac:\ D>0 \text{ distinct real},\ D=0 \text{ equal},\ D<0 \text{ no real zeroes}\]

Common mistake: Writing (a + b)^2 = a^2 + b^2 and forgetting the middle term 2ab.

Exam tips

Check a factorisation by expanding it back. You must get the original polynomial exactly.
If all powers are even, all coefficients are positive and the constant is positive, there are no real zeroes.
To find a remainder quickly, use p(a) instead of doing the full long division.

Beta: made with AI and checked by a second AI. A teacher has not reviewed it yet.

Sample questions

What is the degree of the polynomial 3x^2 + 5x + 6?
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Answer: C. 2
The degree of a polynomial is the highest power of the variable in the polynomial. Here, the highest power of x is 2.
Find the coefficient of x in the polynomial 7x^3 + 4x^2 - x + 5.
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Answer: C. -1
The coefficient of x in the polynomial 7x^3 + 4x^2 - x + 5 is -1.
How many terms are there in the expansion of (x + y)^2?
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Answer: C. 3
The expansion of (x + y)^2 is x^2 + 2xy + y^2, which consists of 3 terms.
Which of the following is a polynomial identity?
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Answer: A. (x + 1)^2 = x^2 + 2x + 1
A polynomial identity is an equation that holds true for all values of the variable. The equality (x + 1)^2 = x^2 + 2x + 1 is true for any value of x.
If p(x) = x^3 - 2x^2 + x - 2, find p(2).
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Answer: A. 0
Substituting x = 2 in p(x), we get p(2) = 2^3 - 2*2^2 + 2 - 2 = 0, verifying x = 2 is a root.

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