Answer:
[tex]3{x}^{3} + 6{x}^{2} + 8x + 24 + \frac{47}{x - 2}[/tex]
Step-by-step explanation:
Since the divisor is in the form of x - c, use what is called Synthetic Division. Remember, in this formula, -c gives you the OPPOSITE terms of what they really are, so do not forget it. Anyway, here is how it is done:
2| 3 0 −4 8 −1
↓ 6 12 16 48
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3 6 8 24 47 → [tex]3{x}^{3} + 6{x}^{2} + 8x + 24 + \frac{47}{x - 2}[/tex]
You start by placing the c in the top left corner, then list all the coefficients of your dividend [3x⁴ - 4x² + 8x - 1]. You bring down the original term closest to c then begin your multiplication. Now depending on what symbol your result is tells you whether the next step is to subtract or add, then you continue this process starting with multiplication all the way up until you reach the end. Now, when the last term is 0, that means you have no remainder. Finally, your quotient is one degree less than your dividend, so that 3 in your quotient can be a 3x³, the 6x² follows right behind it, green then 8x, 24, and finally, your remainder of 47, which gets set over the divisor of [tex]x - 2[/tex], giving you the quotient of [tex]3{x}^{3} + 6{x}^{2} + 8x + 24 + \frac{47}{x - 2}[/tex].
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