Question

Find the remainders when $x^3+3x^2+3x+1$ is divided by (i) $(x+1)$ and (ii) $x-\frac{1}{2}$.

Answer 1

If a polynomial p(x) is divided by x - a then the remainder is p(a).

(i) By putting the value x = - 1 in polynomial p(x), we get
p(x) = p(-1) = $(-1{)}^3+3\times (-1{)}^2+3\times (-1)+1$
p(-1) = -1 + 3 - 3 + 1
p(-1) = 0
Hence by the remainder theorem, 0 is the remainder.

(ii) By putting the value $x=1/2$ in polynomial p(x), we get
p(1/2) = $(1/2{)}^3+3\times (1/2{)}^2+3\times \left(1/2\right)+1$
= 1/8 + 3/4 + 3/2 + 1
= $(1+6+12+8)/8$
= 27/8
Hence by the remainder theorem, 27 / 8 is the remainder.