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Polynomials Quiz - CBSE Class 8 Mathematics

Free practice quiz: Polynomials (CBSE Class 8 Mathematics). 15 multiple-choice questions with answers and explanations. Play online, no sign-up needed.

This CBSE Class 8 Mathematics practice quiz has 15 multiple-choice questions on Polynomials, and 15 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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Parts and Degree of Polynomials

A polynomial adds terms like ax^n, where every power n is a whole number (0, 1, 2, ...).
The degree is the highest power of the variable. The constant term is the term with no variable.
A monomial has one term, a binomial has two terms, and a trinomial has three terms.

General polynomial

\[p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0,\ a_n \neq 0\]

Degree

\[\deg p(x) = n\]

Not polynomials

\[\frac{1}{x},\ \sqrt{x},\ x^{-k}\]

Common mistake: Calling 1/x or √x a polynomial term, or reading the degree from the first term instead of the highest power.

Adding and Multiplying Polynomials

To add polynomials, combine like terms by adding the coefficients of the same powers of x.
To multiply, multiply every term of one polynomial by every term of the other, then combine like terms.

Adding like terms

\[ax^k + bx^k = (a+b)x^k\]

Product of binomials

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

Degree of a product

\[\deg(p \cdot q) = \deg p + \deg q\]

Common mistake: Multiplying degrees instead of adding them, or forgetting the middle term when multiplying two binomials.

Value, Division and Remainder

The value p(a) is found by putting x = a in every term and simplifying carefully.
In long division, divide the leading terms, multiply back, subtract, and repeat until the remainder's degree is smaller.
Remainder theorem: when p(x) is divided by x − a, the remainder equals p(a).

Division algorithm

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

Remainder theorem

\[r = p(a) \text{ when dividing by } (x - a)\]

Common mistake: Skipping a missing power during long division. Write it with coefficient 0 so the columns stay lined up.

Factors and Roots

A root is a value a that makes p(a) = 0. Then (x − a) is a factor.
Factor x^2 + bx + c by finding two numbers whose sum is b and whose product is c.
For four terms, try grouping: take a common factor from pairs, then take out the common bracket.

Factor theorem

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

Polynomial with roots a and b

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

Difference of squares

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

Common mistake: Getting signs wrong. A root at x = a gives the factor (x − a), not (x + a).

Sign of Quadratics, Real Roots

x^2 + bx + c stays positive for all x exactly when it has no real roots, so b^2 − 4c < 0.
Real roots are the x-values where the graph crosses or touches the x-axis.
Squares are never negative. Rewriting an expression as squares plus a positive number shows it has no real roots.

Always positive (leading coefficient 1)

\[x^2 + bx + c > 0\ \forall x \iff b^2 - 4c < 0\]

Square is non-negative

\[(x + k)^2 \geq 0\]

Common mistake: Thinking b > 0 and c > 0 is enough for always positive. What matters is the condition b^2 − 4c < 0.

Exam tips

Check a possible root quickly by substituting it. If the value is zero, it gives a factor.
Arrange terms in descending powers before adding, multiplying or dividing polynomials.
Check a factorisation by multiplying the factors back to get the original polynomial.

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Sample questions

What is the degree of the polynomial 4x^3 - 3x^2 + 2x - 5?
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Answer: A. 3
The degree of a polynomial is the highest power of the variable in the expression. Here, the highest power is 3 from the term 4x^3.
What is the sum of the polynomials (2x^2 + 3x + 4) and (3x^2 - 2x + 1)?
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Answer: A. 5x^2 + x + 5
To sum the polynomials, we combine like terms: (2x^2 + 3x + 4) + (3x^2 - 2x + 1) = (2 + 3)x^2 + (3 - 2)x + (4 + 1) = 5x^2 + x + 5.
What is the value of the polynomial 5x^2 - 3x + 2 when x = 2?
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Answer: C. 14
Substituting x = 2 gives us 5(2^2) - 3(2) + 2 = 5(4) - 6 + 2 = 20 - 6 + 2 = 14.
If p(x) = 2x^3 + 3x^2 - x + 4, what is p(1)?
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Answer: A. 8
Substituting x = 1 into p(x): p(1) = 2(1)^3 + 3(1)^2 - (1) + 4 = 2 + 3 - 1 + 4 = 8.
The polynomial g(x) = x^2 + mx + n is known to be positive for all x. What can you conclude about m and n?
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Answer: C. m^2 - 4n < 0
For the quadratic to be positive for all x, the discriminant must be less than 0, which leads to the inequality m^2 - 4n < 0.