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Polynomials Practice

Solve chapter-level practice questions for Polynomials with reveal-only solutions and quick revision support.

Practice Set 1 — Degree, Zeroes, Remainder Theorem

Identifying polynomials, evaluating, and using the Remainder Theorem.

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Q1.
Which of these are polynomials in one variable? 3x2−2x+53x^2 - 2x + 5, x+1\sqrt{x} + 1, x−2+xx^{-2} + x, 55
Q2.
Write the degree of 7−x3+4x5−2x27 - x^3 + 4x^5 - 2x^2.
Q3.
Find p(0)p(0), p(1)p(1), and p(−2)p(-2) for p(x)=x2−3x+2p(x) = x^2 - 3x + 2.
Q4.
Find the zero of the polynomial p(x)=3x−5p(x) = 3x - 5.
Q5.
Using the Remainder Theorem, find the remainder when x3−2x2+x+1x^3 - 2x^2 + x + 1 is divided by (x−1)(x - 1).
Q6.
Find the value of kk if (x−1)(x - 1) is a factor of p(x)=kx2−3x+kp(x) = kx^2 - 3x + k.

Practice Set 2 — Factorisation and Identities

Splitting the middle term, the Factor Theorem for cubics, and identities.

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Q1.
Factorise x2+7x+12x^2 + 7x + 12.
Q2.
Factorise 6x2−5x−66x^2 - 5x - 6 by splitting the middle term.
Q3.
Factorise x3−23x2+142x−120x^3 - 23x^2 + 142x - 120 given that (x−1)(x - 1) is a factor.
Q4.
Expand (2a−3b)2(2a - 3b)^2.
Q5.
Evaluate 104×96104 \times 96 using an identity.
Q6.
If x+y+z=0x + y + z = 0, show that x3+y3+z3=3xyzx^3 + y^3 + z^3 = 3xyz.

Quick Q&A Before You Revise

Is $\sqrt{2}x + 3$ a polynomial?

Yes. The coefficient can be any real number, including an irrational number like √2. What matters is that the power of the variable (here 1) is a whole number.

What is the difference between a zero of a polynomial and the value of a polynomial?

The value of p(x) at x = a is simply p(a). A zero is a special value of x for which p(a) = 0.

How does the Factor Theorem help in factorising a cubic polynomial?

Try small integer values (factors of the constant term). If p(a) = 0, then (x − a) is a factor. Divide the cubic by (x − a) to get a quadratic, which can then be factorised by splitting the middle term.

How many zeroes can a polynomial of degree n have?

At most n real zeroes. A linear polynomial has exactly one; a quadratic has at most two; a cubic has at most three.

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