In each of the following, using the remainder Theorem, find the remainder when f(x) is divided by g(x) and verify the result y actual division.

f(x) = $9 x^{3} - 3 x^{2}$ + x -5, g(x) = x - $\frac{2}{3}$

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Solution

f(x) = $9 x^{3} - 3 x^{2}$ + x -5, g(x) = x - $\frac{2}{3}$

$x - \frac{2}{3} = 0, x = \frac{2}{3}$

$∴ R e m a i n d e r f (\frac{2}{3}) = 9 {(\frac{2}{3})}^{3} - 3 {(\frac{2}{3})}^{2} + \frac{2}{3} - 5$

$= 9 \times \frac{8}{27} - 3 \times \frac{4}{9} + \frac{2}{3} - 5$

$= \frac{8}{3} - \frac{4}{3} + \frac{2}{3} - \frac{5}{1}$

$= \frac{8 - 4 + 2 - 15}{3} = \frac{10 - 19}{3} = \frac{- 9}{3} = - 3$

$∴$ Remainder = - 3

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In each of the following, using the remainder Theorem, find the remainder when f(x) is divided by g(x) and verify the result y actual division. f(x) = 9x3−3x2 + x -5, g(x) = x -23

f(x) = 9x3−3x2 + x -5, g(x) = x -23 x−23=0,x=23 ∴Remainder f(23)=9(23)3−3(23)2+23−5 =9×827−3×49+23−5 =83−43+23−51 =8−4+2−153=10−193=−93=−3 ∴ Remainder = - 3

In each of the following, using the remainder Theorem, find the remainder when f(x) is divided by g(x) and verify the result y actual division.

f(x) = $9 x^{3} - 3 x^{2}$ + x -5, g(x) = x - $\frac{2}{3}$