Question

• What is the reminder? when x⁴- 3x²- 5x is divided by 2x-1

Answer 1

$$\dfrac{x^4- 3x^2- 5x}{2x-1}$$

$$=\dfrac{1}{2}x^3+\dfrac{x^4-x^4+\dfrac{1}{2}x^3- 3x^2- 5x}{2x-1}$$

$$=\dfrac{1}{2}x^3+\dfrac{1}{4}x^2+\dfrac{\dfrac{1}{2}x^3-\dfrac{1}{2}x^3+\dfrac{1}{4}x^2- 3x^2- 5x}{2x-1}$$

$$=\dfrac{1}{2}x^3+\dfrac{1}{4}x^2-\dfrac{11}{8}x+\dfrac{-\dfrac{11}{4}x^2+\dfrac{11}{4}x^2-\dfrac{11}{8}x-5x}{2x-1}$$

$$=\dfrac{1}{2}x^3+\dfrac{1}{4}x^2-\dfrac{11}{8}x-\dfrac{51}{16}+\dfrac{-\dfrac{51}{8}x+\dfrac{51}{8}x-\dfrac{51}{16}}{2x-1}$$

$$=\dfrac{1}{2}x^3+\dfrac{1}{4}x^2-\dfrac{11}{8}x-\dfrac{51}{16}+\dfrac{-\dfrac{51}{16}}{2x-1}$$

The reminder when $x^4- 3x^2- 5x$ is divided by $2x-1$ is $-\dfrac{51}{16}.$