What Mistake Was Made in This Polynomial Long Division?

In summary, using polynomial long division, the fraction \frac{x^3 + 4x^2 + 3}{x+4} can be rewritten as x^2 + \frac{3}{x+4}.
  • #1
shamieh
539
0
rewrite using polynomial long division

\(\displaystyle \frac{x^3 + 4x^2 + 3}{x+4}\)

so I did \(\displaystyle x+4 \sqrt{x^3 + 4x^2 + 0x + 3}\)

and got \(\displaystyle x^2 + 1 - \frac{1}{x+4}\)

What am i doing wrong? How is that incorrect?
 
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  • #2
I think you made an error in your first subtraction:

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

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


\(\displaystyle \begin{array}{cccccccccc}
&&&& x^2 & + & 0x & + & 0 \\
&& --&--&--&--&--&--&-- \\
x+4 & ) & x^3 & + & 4x^2 &+&0x& + & 3 \\
&& x^3 & + & 4x^2 \\
&& --&--&-- \\
&&&& 0x^2 &+& 0x \\
&&&& 0x^2 &+& 0x \\
&&&& --&--&--&--\\
&&&&&& 0x &+& 3 \\
&&&&&& 0x &+& 0 \\
&&&&&& --&--&-- \\
&&&&&&&& 3 \end{array}\)
Answer: \(\displaystyle \;x^2 + \frac{3}{x+4}\)
 

FAQ: What Mistake Was Made in This Polynomial Long Division?

What is polynomial long division?

Polynomial long division is a method used to divide one polynomial by another polynomial in order to find the quotient and remainder. It is similar to long division with numbers, but instead of dividing numbers, we are dividing polynomials.

When is polynomial long division used?

Polynomial long division is used when we need to simplify complex algebraic expressions, solve polynomial equations, or find the partial fraction decomposition of a rational expression. It is also used in calculus to find limits and derivatives of rational functions.

What are the steps for polynomial long division?

The steps for polynomial long division are as follows:

  • Step 1: Arrange the polynomial in descending order of exponents.
  • Step 2: Divide the first term of the dividend (numerator) by the first term of the divisor (denominator) to get the first term of the quotient.
  • Step 3: Multiply the first term of the quotient by the entire divisor and subtract it from the dividend.
  • Step 4: Bring down the next term of the dividend to the remainder.
  • Step 5: Repeat steps 2-4 until there are no more terms left in the dividend.

What is the difference between synthetic division and polynomial long division?

The main difference between synthetic division and polynomial long division is that synthetic division can only be used when the divisor is of the form x – a, while polynomial long division can be used for any type of polynomial divisor. Additionally, synthetic division is a quicker method, but it can only be used to find the quotient and not the remainder.

How do I know if I have correctly done polynomial long division?

To check if you have correctly done polynomial long division, you can do the following:

  • Multiply the quotient by the divisor and add the remainder to the result.
  • The resulting polynomial should be equivalent to the original dividend.
  • If the resulting polynomial is not the same, then you have made a mistake and should redo the division.

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