For a fixed positive integer

MATHEMATICS · Subjective · CBSE · Mathematics · Polynomials · Polynomials

Problem

n

, consider the expression

S_n(x)=\frac{x^{n+1}-1}{x-1},\;x\ne 1.

(i) Show that for

x\ne 1

S_n(x)=x^{n}+x^{n-1}+\dots+x+1.

(ii) Hence or otherwise, explain why

S_n(x)

represents a polynomial in

x

of degree

n

except possibly at

x=1

. (iii) Determine the value that must be assigned to

S_n(x)

x=1

so that the resulting function becomes a polynomial defined on the entire real line. Prove that your choice makes the function continuous at

x=1

. (iv) Taking

n=4

, write the explicit polynomial obtained in part (iii) and state its degree, leading coefficient and constant term.