1.

Find the remainder when `x^3+3x^2+3x+1`is divided by(i) `x+1` (ii) `x-1/2` (iii) `x` (iv) `x+pi` (v) `5+2x`

Answer» From remainder theorem, we know that if `P(x)` is divided by `(x-a)`, then, ` P(a)` will be the remainder.
In this question,
`P(x) = x^3+3x^2+3x+1`

(i) Here, `a = -1`
`P(-1) = (-1)^3+3(-1)^2+3(-1)+1 = -1+3-3+1=0`
Remainder is `0`.

(ii) Here, `a = 1/2`
`P(1/2) = (1/2)^3+3(1/2)^2+3(1/2)+1 =1/8+3/4+3/2+1=27/8`
Remainder is `27/8`.

(iii) Here, `a = 0`
`P(0) = (0)^3+3(0)^2+3(0)+1 =1`
Remainder is `1`.

(iv)Here, `a = -pi`
`P(-pi) = (-pi)^3+3(-pi)^2+3(-pi)+1 =-pi^3+3pi^2-3pi+1`
Remainder is `-pi^3+3pi^2-3pi+1`.

(v)Here, `a = -5/2`
`P(-5/2) = (-5/2)^3+3(-5/2)^2+3(-5/2)+1 =-125/8+75/4-15/2+1`
` =1/8(-125+150-60+8=-27/8`
Remainder is `-27/8`.


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