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Polynomials Practice

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

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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? $3x^2 - 2x + 5$, $\sqrt{x} + 1$, $x^{-2} + x$, $5$
Q2. Write the degree of $7 - x^3 + 4x^5 - 2x^2$.
Q3. Find $p(0)$, $p(1)$, and $p(-2)$ for $p(x) = x^2 - 3x + 2$.
Q4. Find the zero of the polynomial $p(x) = 3x - 5$.
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Q5. Using the Remainder Theorem, find the remainder when $x^3 - 2x^2 + x + 1$ is divided by $(x - 1)$.
Q6. Find the value of $k$ if $(x - 1)$ is a factor of $p(x) = kx^2 - 3x + k$.

Practice Set 2 — Factorisation and Identities

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

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

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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