Question

In each of the following, using the remainder theorem, find the remainder when f(x) is divided by g(x):
f(x) = 3x⁴+2x³ x²/3 - x/9 + 2/27, g(x) = x+ 2/3

Answer 1

Given : f(x) = 3x⁴ + 2x³ - x²/3 - x/9 + 2/27, g(x) = x + 2/3

By remainder theorem,  when f(x) is divided by g(x)  = x + ⅔ , the remainder is equal to f(-⅔) :  

Now, f(x) = 3x⁴ + 2x³ - x²/3 - x/9 + 2/27

f(- ⅔ ) = 3(- ⅔)⁴ + 2(- ⅔)³ - (- ⅔)² /3 - (- ⅔) /9 + 2/27

f(- ⅔ ) = 3 × 16/81 + 2 × -8/27 - 4/9 × ⅓ + ⅔ × 1/9 + 2/27

f(- ⅔ ) = 16/27 - 16/27 - 4/27 + 2/27 + 2/27

f(- ⅔ ) = - 4/27 + (2/27 + 2/27)

f(- ⅔ ) = - 4/27 + (2 + 2)/27

f(- ⅔ ) = - 4/27 + 4/27

f(- ⅔ ) = 0

Hence, the remainder when f(x) is divided by g(x) is 0.

 

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Answer 2

$\bf \red{solution}$

• g(x) =>x + 2/3=0

=>x=-2/3

f(x) = 3x⁴ + 2x³ - x²/3 - x/9 + 2/27

putting the value of x

f(- ⅔ ) = 3(- ⅔)⁴ + 2(- ⅔)³ - (- ⅔)² /3 - (- ⅔) /9 + 2/27

f(- ⅔ ) = 3 × 16/81 + 2 × -8/27 - 4/9 × ⅓ + ⅔ × 1/9 + 2/27

f(- ⅔ ) = 16/27 - 16/27 - 4/27 + 2/27 + 2/27

f(- ⅔ ) = - 4/27 + (2/27 + 2/27)

f(- ⅔ ) = - 4/27 + (2 + 2)/27

f(- ⅔ ) = - 4/27 + 4/27

f(- ⅔ ) = 0

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