How to divide polynomials - Algebra

First, set up the division as the following:

Look at the leading term in the divisor and in the dividend. Divide by gives ; therefore, put on the top:

Then take that and multiply it by the divisor, , to get . Place that under the division sign:

Subtract the dividend by that same and place the result at the bottom. The new result is , which is the new dividend.

Now, is the new leading term of the dividend. Dividing by gives 5. Therefore, put 5 on top:

Multiply that 5 by the divisor and place the result, , at the bottom:

Perform the usual subtraction:

Therefore the answer is with a remainder of , or .

First we will factor the numerator:

Then factor the denominator:

We can re-write the original fraction with these factors and then cancel an (x-5) term from both parts:

When dividing polynomials, subtract the exponent of the variable in the numberator by the exponent of the same variable in the denominator.

If the power is negative, move the variable to the denominator instead.

First move the negative power in the numerator to the denominator:

Then subtract the powers of the like variables:

The numerator is equivalent to

The denominator is equivalent to

Dividing the numerator by the denominator, one gets

First let us find a common denominator as follows:

Now we can subtract the numerators which gives us :

So the final answer is

Factor both the numerator and the denominator which gives us the following:

After cancelling we get

We can accomplish this division by re-writing the problem as a fraction.

The denominator will distribute, allowing us to address each element separately.

Now we can cancel common factors to find our answer.

7 in the denominator is a common factor of the three coefficients in the numerator, which allows you to divide out the 7 from the denominator:

Then divide by :

We are dividing the polynomial by a monomial. In essence, we are dividing each term of the polynomial by the monomial. First I like to re-write this expression as a fraction. So,

So now we see the three terms to be divided on top. We will divide each term by the monomial on the bottom. To show this better, we can rewrite the equation.

Now we must remember the rule for dividing variable exponents. The rule is So, we can use this rule and apply it to our expression above. Then,

When dividing exponents, you subtract exponents that share the same base, so

and and .

Do not forget to "add the opposite" when subtracting negative numbers).

Now, you have

But you are not done yet! Remember, you do not want to have a negative exponent, and the way to turn the negative exponent into a positive exponent is to take its reciprocal, like this:

You keep the rest of the equation in the numerator, leaving you with

4 goes into 24, 12, 8, and 4.

Similarly, the smallest exponent of x in the four terms is 2, and the smallest exponent of y in the four terms is 1.

Hence the GCF must be .

Divide the leading coefficients to get the first term of the quotient:

, the first term of the quotient

Multiply this term by the divisor, and subtract the product from the dividend:

Repeat these steps with the differences until the difference is an integer. As it turns out, we need to repeat only once:

, the second term of the quotient

, the remainder

Putting it all together, the quotient can be written as .

Divide each of the terms in the numerator by the denominator:

Simplify each term above to get the final:

First, rewrite this problem so that the missing term is replaced by

Divide the leading coefficients:

, the first term of the quotient

Multiply this term by the divisor, and subtract the product from the dividend:

Repeat this process with each difference:

, the second term of the quotient

One more time:

, the third term of the quotient

, the remainder

The quotient is and the remainder is ; this can be rewritten as a quotient of

The numerator can be factored into

,

which when divided by ,

gives us .

Alternate method: Long division of the numerator by the denominator gives the same answer.

When we divide a polynomial by another polynomial we get:

1. Quotient
2. Remainder (if one exists)

In our problem the long division results in:

1. A quotient of
2. A remainder of

This can easily be solved by factoring using the difference of cubes formula:

First, convert the given polynomial into a difference of two cubes:

Compare this with the difference of cubes formula above to get:

By dividing the above numerator by the given denominator we get:

To simplify, we must use exponent rules. For exponents in fractions, we can subtract the exponent of the denominator from the exponent in the numerator.

With this rule, we can rewrite the problem.

Remember that negative exponents get moved back to the denominator, turning them positive.

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