Polynomials – CBSE Class 10 Mathematics Notes & Guide

Polynomials form a major part of Class 10 mathematics. They are expressions involving powers of a variable with constant coefficients. Understanding their degree, zeroes, factors and the theorems that connect them helps in solving equations quickly and accurately.

Key concepts

Degree and coefficients

The degree of a polynomial is the highest power of the variable that appears with a non‑zero coefficient. The numbers multiplying each power are called coefficients. For example, in 3x³‑5x²+2x‑7, the degree is 3 and the coefficients are 3, –5, 2 and –7 respectively.

Zeroes and factors

A zero (or root) of a polynomial f(x) is a number α such that f(α)=0. If α is a zero, then (x‑α) is a factor of the polynomial. Conversely, every linear factor (x‑α) gives a zero α.

Remainder theorem

When a polynomial f(x) is divided by x‑a, the remainder is the value f(a). This allows us to find the remainder without performing long division.

Factor theorem

A special case of the remainder theorem: x‑a is a factor of f(x) iff f(a)=0. Hence, checking whether a number is a zero tells us if the corresponding linear factor divides the polynomial exactly.

Relation between zeroes and coefficients

For a polynomial of degree n written as anxⁿ + a(n-1)xⁿ⁻¹+… + a_1x + a₀ with zeroes α₁, α₂, … , αₙ (including repeated zeroes), the following relationships hold:

• Sum of zeroes = -a_(n-1)/aₙ
• Sum of product of zeroes taken two at a time = a_(n-2)/aₙ
• …
• Product of all zeroes = (-1)ⁿ a₀ / aₙ

These formulas are derived from expanding aₙ(x‑α₁)(x‑α₂)…(x‑αₙ).

Important definitions and formulas

• Degree: highest exponent of x with non‑zero coefficient.
• Coefficient: constant multiplying a power of x.
• Zero (root): value α such that f(α)=0.
• Factor: polynomial g(x) that divides f(x) exactly, leaving remainder 0.
• Remainder theorem: remainder on dividing f(x) by x‑a = f(a).
• Factor theorem: x‑a is a factor of f(x) ⇔ f(a)=0.
• Relations for a cubic ax³+bx²+cx+d:
• α+β+γ = -b/a
• αβ+βγ+γα = c/a
• αβγ = -d/a
• General relation for degree n as given in the key concepts section.

Solved examples

Example 1: Finding remainder using the remainder theorem

Find the remainder when f(x)=2x⁴‑3x³+5x‑6 is divided by x‑2. Solution

1. According to the remainder theorem, remainder = f(2).
2. Substitute x=2:

• 2(2)⁴ = 2·16 = 32
• -3(2)³ = -3·8 = -24
• +5·2 = 10
• -6 remains.

1. Add: 32‑24+10‑6 = 12.

Remainder = 12.

Example 2: Using factor theorem to test a zero

Check whether x+1 is a factor of g(x)=x³‑4x²‑7x+10. Solution

1. Factor theorem says x+1 (i.e., x‑(-1)) is a factor if g(-1)=0.
2. Compute g(-1):

• (-1)³ = -1
• -4(-1)² = -4·1 = -4
• -7(-1) = +7
• +10 remains.

1. Add: -1‑4+7+10 = 12.

Since g(-1) = 12 ≠ 0, x+1 is not a factor.

Example 3: Relation between zeroes and coefficients

The cubic polynomial p(x)=2x³‑3x²‑8x+12 has zeroes α,β,γ. Find α+β+γ and αβγ. Solution

1. Write the polynomial in standard form ax³+bx²+cx+d with a=2, b=-3, c=-8, d=12.
2. Sum of zeroes = -b/a = -(-3)/2 = 3/2.
3. Product of zeroes = (-1)³ d/a = -d/a = -12/2 = -6.

Therefore, α+β+γ = 1.5 and αβγ = -6.

Common mistakes to avoid

• Confusing degree with number of terms – degree depends only on the highest power, not on how many terms are present.
• Ignoring sign while applying the factor theorem – for factor (x‑a) we must evaluate the polynomial at x = a, not -a.
• Leaving out the leading coefficient in the zero‑coefficient relations – the formulas involve division by the leading coefficient aₙ.
• Treating repeated zeroes incorrectly – if a zero repeats, it contributes multiple times to the sum and product formulas.
• Using the remainder theorem on a divisor that is not linear – the theorem works only for divisors of the form x‑a.

Exam tips

• Memorise the remainder and factor theorems; they save time in long‑division questions.
• When asked for zeroes, first check easy integer candidates using the factor theorem before attempting factorisation.
• Write the polynomial in the form aₙ(x‑α₁)(x‑α₂)… to quickly derive relations between zeroes and coefficients.
• In multiple‑choice questions, substitute the given value directly into the polynomial to verify a zero; no need for full factorisation.
• Keep a small table of common cubic relations handy for quick reference.

Frequently asked questions

What is the difference between a zero and a factor?

A zero (root) is a number α that makes the polynomial zero, i.e., f(α)=0. The corresponding factor is the linear polynomial (x‑α). Every zero gives a factor, and every linear factor gives a zero.

Can the remainder theorem be used for a divisor like x²‑4?

No. The remainder theorem is valid only for linear divisors of the form x‑a. For higher‑degree divisors we must perform polynomial division or use the factor theorem repeatedly.

How do I find the sum of zeroes for a quadratic equation?

For a quadratic ax²+bx+c, the sum of its zeroes is -b/a and the product is c/a. This follows directly from the general relations for degree 2.