Simplifying Polynomials - Algebra II

First, factor the numerator of the quotient term by recognizing the difference of squares:

Cancel out the common term from the numerator and denominator:

FOIL (First Outer Inner Last) the first two terms of the equation:

Combine like terms:

First distribute the :

Then distribute the :

Finally combine like terms:

You can look at this as the sum of two expressions multiplied by the difference of the same two expressions. Use the pattern

,

where and .

To find , you use the formula for perfect squares:

,

where and .

Substituting above, the final answer is .

Use the distributive property to obtain each term:

This is not a FOIL problem, as we are adding rather than multiplying the terms in parenteses.

Add like terms to solve.

Combining these terms into an expression gives us our answer.

In simplifying this expression, be mindful of the order of operations (parenthical, division/multiplication, addition/subtraction).

Since operations invlovling parentheses occur first, distribute the factors into the parenthetical binomials. Note that the outside the first parenthetical binomial is treated as since the parenthetical has a negative (minus) sign in front of it. Similarly, multiply the members of the expression in the second parenthetical by because of the negative (minus) sign in front of it. Distributing these factors results in the following polynomial.

Now like terms can be added and subtracted. Arranging the members of the polynomial into groups of like terms can help with this. Be sure to retain any negative signs when rearranging the terms.

Adding and subtracting these terms results in the simplified expression below.

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.

Since we are only adding and subtracting (there is no multiplication or division), we can remove the parentheses.

Regroup the expression so that like variables are together. Remember to carry positive and negative signs.

For all fractional terms, find the least common multiple in order to add and subtract the fractions.

Combine like terms and simplify.

Use FOIL to expand the monomials.

Return this expansion to the original expression.

Distribute negative sign.

Combine like terms.

Distribute the negative sign through all terms in the parentheses:

Add the second half of the expression, to get:

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 .

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

To simplify you combind like terms:

Answer:

Simplify

Distribute the negative:

Then combinde like terms

Answer:

First, we will need to distribute the minus sign.

Then, combine like terms.

Replace: :

When combining polynomials, only combine like terms. With the like terms, combine the coefficients. Your answer is .