What Mistake Was Made in This Polynomial Long Division?

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SUMMARY

The discussion centers on the polynomial long division of the expression $$\frac{x^3 + 4x^2 + 3}{x+4}$$. The correct result of this division is $$x^2 + \frac{3}{x+4}$$, which was miscalculated by the user as $$x^2 + 1 - \frac{1}{x+4}$$. The error occurred during the subtraction step where the user incorrectly simplified the terms, leading to an inaccurate remainder. The proper execution of polynomial long division reveals that the remainder is 3, not 1.

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shamieh
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rewrite using polynomial long division

$$\frac{x^3 + 4x^2 + 3}{x+4}$$

so I did $$x+4 \sqrt{x^3 + 4x^2 + 0x + 3}$$

and got $$x^2 + 1 - \frac{1}{x+4}$$

What am i doing wrong? How is that incorrect?
 
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I think you made an error in your first subtraction:

$x^3 + 4x^2 + 3 - (x + 4)x^2 = x^3 + 4x^2 + 3 - x^3 - 4x^2 = 3$.
 
Hello, shamieh!

Rewrite using polynomial long division: $$ \;\;\frac{x^3 + 4x^2 + 3}{x+4}$$

[math]\begin{array}{cccccccccc}
&&&& x^2 & + & 0x & + & 0 \\
&& --&--&--&--&--&--&-- \\
x+4 & ) & x^3 & + & 4x^2 &+&0x& + & 3 \\
&& x^3 & + & 4x^2 \\
&& --&--&-- \\
&&&& 0x^2 &+& 0x \\
&&&& 0x^2 &+& 0x \\
&&&& --&--&--&--\\
&&&&&& 0x &+& 3 \\
&&&&&& 0x &+& 0 \\
&&&&&& --&--&-- \\
&&&&&&&& 3 \end{array}[/math]
Answer: [math]\;x^2 + \frac{3}{x+4}[/math]
 

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