Technical and student resources

ILLUSTRATIVE EXAMPLES:

Adding and subtracting products of polynomials.

EXERCISES: Adding and subtracting products of polynomials

Simplify each of the following:

Dividing a monomial by another monomial. We have already learned to divide one monomial by another.

we write the quotient as the product of these three factors, — 3xy⁴. This work may be arranged in either of the two ways shown.

Dividing a polonomial by a monomial.

Let us first consider a problem from arithmetic,

Here we can complete the indicated addition in the numerator to get the fraction 45/5, which equals 9. In algebra, however, where we are dealing with literal numbers, the indicated addition or subtraction of terms cannot always be completed; hence we shall consider an alternate method of solving the problem. Thus in the above problem when 10 is divided by 5, the quotient is 2; and when 35 is divided by 5, the quotient is 7. This is expressed in the form:

You will remember that

may also be indicated as (10 + 35) % 5.

The use of parentheses here indicates that each term within the parentheses is to be divided by 5. Thus:

ILLUSTRATIVE EXAMPLES:

Dividing a polynomial by a monomial

Solution.

Applying the above rule, we divide each term of the binomial in the numerator by the monomial in the denominator. This work may be arranged in either of the two forms shown below:

Solution. The work may be arranged in either of the two forms shown below:

EXERCISES: Dividing a polynomial by a monomial

Divide each of the following as indicated

A polynomial divided by a polynomial.

We may divide one polynomial by another polynomial in a manner similar to the long division process in arithmetic. Let us consider the example (3x² - x - 10) /(x — 2); at the same time let us observe the division 832 / 32 in order that we may compare the two processes.



ILLUSTRATIVE EXAMPLES : Dividing a polynomial by a polynomial.

Note: In each of the illustrative examples the terms of both the divisor and the dividend were arranged in the order of descending powers of the same letter.

To divide one polynomial by another:

1. Arrange the terms of both the dividend and the divisor according to the descending (or ascending) powers of the same letter. lf more than one letter appears in both dividend and divisor, select one letter and arrange as explained above.

2. Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient.

3. Multiply each term of the divisor by the first term of the quotient and then subtract their products from the given dividend to obtain a second dividend.

4. Follow the pattern of step 2 using the second dividend to obtain the second term of the quotient.

5. Repeat the patterns of steps 2 and 3 until the division is complete.