Polynomial Functions - College Algebra

The rules for adding fractions containing unknowns are the same as for fractions containing explicit numbers, so you can guide yourself by recalling how you would proceed adding fractions such as,

As you know you need to write them with a common denominator. In this case the least common denominator is . So simply multiply the numerator and denominator of each fraction by the denominator of the other fraciton.

Notice that and are equal to one, this ensures that we are not changing the value of the fractions, we are changing only the representation of the value.

Similairily, the procedure for an algebraic expression containing unknowns parallels this idea,

Now we can add the numerators directly since we now have both terms expressed with a common denominator, .

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:

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.

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

Simplify the following expression:

First, let's multiply the 3x through:

Next, divide out the x from the bottom:

So our answer is:

Simplify the following expression:

To begin, we need to recognize the bottom as a difference of squares. Rewrite it as follows.

So our answer is: