OPERATIONS WITH POLYNOMIALS
Adding and subtracting polynomials is simply the adding and subtracting of their liketerms. There is a great similarity between the operations with polynomials and denominate numbers. Compare the following examples:
1. Add 5 qt and 1 pt to 3 qt and 2 pt.
2.Add 5x + y to 3x + 2y.
One method of adding polynomials (shown inthe above examples) is to place like terms in columns and to find the algebraic sum of the like terms. For example, to add 3a + b - 3c, 3b + c - d, and 2a + 4d, we would arrange the polynomials as follows:
Subtraction may be performed by using thesame arrangement-that is, by placing terms of the subtrahend under the like terms of the minuend and carrying out the subtraction with due regard for sign. Remember, in subtraction the signs of all the terms of the subtrahend must first be mentally changed and then the process completed as in addition. For example, subtract 10a + b from 8a - 2b, as follows:
Again, note the similarity between this type ofsubtraction and the subtraction of denominate numbers.
Addition and subtraction of polynomials alsocan be indicated with the aid of symbols of grouping. The rule regarding changes of sign when removing parentheses preceded by a minus sign automatically takes care of subtraction.
For example, to subtract 10a + b from 8a - 2b,we can use the following arrangement:
Similarly, to add -3x + 2y to -4x - 5y, we canwrite
Practice problems. Add asindicated, in each of the following problems:
In problems 5 through 8, perform the indicated operations and combine like terms.
5. (2a + b) - (3a + 5b)
MULTIPLICATION OF A POLYNOMIAL BY A MONOMIAL
We can explain the multiplication of a polynomial by a monomial by using an arithmeticexample. Let it be required to multiply the binomial expression, 7 - 2, by 4. We may write this 4 x (7 - 2)orsimply 4(7 - 2). Now 7 - 2 = 5. Therefore, 4(7 - 2) = 4(5) = 20. Now, let us then subtract. Thus, 4(7 - 2) = (4 x 7) - (4 x 2) = 20. Both methods give the same result. The second method makes use of the distributive law of multiplication.
When there are literal parts in the expression to be multiplied, the first method cannotbe used and the distributive method must be employed. This is illustrated in the following examples:
Thus, to multiply a polynomial by a monomial,multiply each term of the polynomial by the monomial.
Practice problems. Multiply as indicated: