Polynomial rings

Objective. Use division, irreducibility and factorisation.

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2. Worked example
3. Practice
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Learn: the key idea

The polynomial ring R[x] over a ring R consists of formal sums Σaᵢxⁱ with coefficients in R, where addition and multiplication follow the usual polynomial rules; when R is a field, R[x] supports division with remainder analogous to integer division. A polynomial is irreducible over a field if it cannot be factored into two polynomials of lower positive degree in that field, and unique factorisation into irreducibles holds in F[x] for a field F, just as primes factorise integers. For example, x²+1 is irreducible over ℝ (no real roots) but factors as (x−i)(x+i) over ℂ, showing irreducibility depends on the base field.

Worked example

Divide the polynomial x³ − 2x² + x − 2 by x − 2 using polynomial division.

1 $Divide leading terms: x³ \div x = x²; multiply and subtract: x³-2x²+x-2 - x²(x-2) = x-2$
2 $Divide leading terms: x \div x = 1; multiply and subtract: (x-2) - 1(x-2) = 0$
3 $Quotient terms collected: x² + 0x + 1$
4 $Quotient is x² + 1 with remainder 0$

Answer: x² + 1

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Common mistake. Students assume a polynomial with no rational roots must be irreducible over ℚ, but a degree-4 polynomial can factor into two irreducible quadratics with no rational roots itself, such as x⁴+4 = (x²-2x+2)(x²+2x+2), which has no rational roots yet is reducible.

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