How do I correctly solve this polynomial long division problem?

In summary, the problem is that you forgot to change the sign of the last term in the dividend, x^6+6x^3-2x^5-7x^2-4x+6. This caused problems with your first partial division and caused you to be unable to solve the problem.
  • #1
Kys91
12
0
I am trying to solve:

[tex]\frac{x^6+6x^3-2x^5-7x^2-4x+6}{x^4-3x^2+2}[/tex]

Using the polynomial long division algorithm.

I order first the terms of the divident, and leave one blank space between -2x^5 and +6x^3

My problem is, I first put x^2 to the quotient, so I get x^4 * x^2 = x^6, but then I multiply x^2 * -3x^2 = -3x^4, which can't be subtracted with -2x^5.

I have tried playing around but with no success.

Thanks
 
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  • #2
You also need to leave blank spaces in your divisor. Better yet, where there should be blank spaces, put the missing term with a coefficient of 0. In your divisor, put 0x3. Multiply that by x2 and you get 0x5 which should be easy to subtract from -2x5. Try doing that with all missing terms in the dividend and divisor because it can be easy to miss something with all those terms.
 
  • #3
Your dividend takes the form as x^6 -2*x^5 +0*x^4 +6*x^3 -7*x^2 -4x +6
Restating your divisor as x^4 +0*x^3 -3*x^2 +0*x +2

Notice every degree of x must be shown. This is like keeping "place value" in "integers" were those to be divided, but now we are keeping track of powers of x, not powers of 10.

Your first partial division was (x^6)/(x^4)=x^2, this was good. Now, how much complete divisor do you subtract? Find this by performing multiplication:
(x^2)*(x^4 +0*x^3 -3*x^2 +0*x +2)=x^6 +0*x^5 -3*x^4 +0*x^3 +2*x^2

Now you write x^6 +0*x^5 -3*x^4 +0*x^3 +2*x^2 in proper alignment under the dividend and perform your first subtraction. Now, continue from here.
 
  • #4
Thank you, much easier with putting zeros.

I also noticed that I messed up a lot doing this: 0x^4 - (-3x^4) = +3x^4, I forgot to change the sign many times.

Your help is very much appreciated.
 

What is polynomial long division?

Polynomial long division is a method used to divide two polynomials, which are algebraic expressions made up of variables and coefficients. It is similar to long division with numbers, but involves dividing terms with variables instead.

Why is polynomial long division useful?

Polynomial long division is useful because it allows us to divide polynomials and obtain the quotient and remainder. This is important in simplifying expressions, finding roots of polynomials, and solving equations.

How do you perform polynomial long division?

To perform polynomial long division, you first arrange the terms of the dividend (the polynomial being divided) and the divisor (the polynomial dividing the dividend) in descending order of degree. Next, you divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. Then, you multiply this term by the entire divisor and subtract it from the dividend. This process is repeated until all terms are divided and a remainder is obtained, if any.

What is the role of coefficients in polynomial long division?

Coefficients are the numbers that appear before the variables in a polynomial. In polynomial long division, the coefficients are used to determine the terms of the quotient. When dividing, the quotient is obtained by dividing the coefficients of the terms with the same degree.

What are some common mistakes to avoid in polynomial long division?

Some common mistakes to avoid in polynomial long division include forgetting to write the placeholders (0x) for missing terms, incorrectly dividing the coefficients, and not properly subtracting the terms in each step. It is also important to double check the final answer by multiplying the quotient and divisor and adding the remainder to ensure it equals the original dividend.

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