If p(x) = x² − (√2 + √3)x + √6, what are the zeroes?

√2 and √3

Answer: A, √2 and √3. For a monic quadratic, x² − (α + β)x + αβ factors as (x − α)(x − β). Here take α = √2 and β = √3. Their sum is α + β = √2 + √3, exactly the quantity in the middle coefficient. Their product is αβ = √2·√3 = √6, exactly the constant term. Therefore p(x) = (x − √2)(x − √3). A zero is a value of x that makes the polynomial equal to zero, so either factor must be zero: x − √2 = 0 gives x = √2, and x − √3 = 0 gives x = √3. B incorrectly treats the product as one zero and introduces 1 without justification. C mistakes the sum of the zeroes for a zero and also adds 0. D changes both signs; its sum would be negative and would not match the polynomial. Memory cue: in x² − Sx + P, the zeroes have sum S and product P.

This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.