Elementary Algebra and Geometry for Schools

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 HOME : Mathematics - Geometry   Addition and Substraction of Polynomials A review of certain definitions and rules. In previous pages you learned that a term is an algebraic expresion which is not separated into-parts by + or signs. You also learned that an algebraic expression containing two terms is a binomial; an algebraic expression containing three terms is a trinomial; and an algebraic expression containing more than a single term is a polynomial. You also learned that like terms, or similar terms, are terms that have identical literal parts. To add or subtract like terms, you add or subtract the numerical coefficients of the like terms and multiply this number by the common literal part of the terms. Unlike terms cannot be collected (added or subtracted); the addition or subtraction of unlike terms can be indicated only. REVIEW EXERCISES In exercises 1-12 find each sum by applying the rule for the addition of signed numbers. In exercises 13-24 perform each subtraction by applying the rule for the subtraction of signed numbers. Answers : Addition of polynomials. Polynomials may be added using procedures we have already learned. If we are to add 3a + 5b -4 and 4a -3b -2 we indicate the addition in this manner: (3a + 5b -4) + (4a -3b -2). Removing parentheses            3a + 5b -4 + 4a -3b -2 Collecting like terms                7a + 2b -6 Answer We can add the polynomials 3a + 5b -4 and 4a -3b -2 more readily y arranging them vertically. We write like terms in the same column since like terms can be combined. Observe that the addition of polynomials consists merely of doing several additions of monomials. EXERCISES: Adding polynomials Find the sum of each of the following: Answers: ILLUSTRATIVE EXAMPLES: Addition of polynomials   1. Find the sum of 3x2+7x -4, 4x -x2+2, and 3 -11x +2x2 Solution. First we arrange the terms of each of the polynomials in the same order. The usual order of arrangement is in descending powers of x; that is, the term containing x2 first, then the term containing x, and then the number. Thus we have: 3x2+7x -4; -x2+ 4x +2; 2x2 -11x + 3. Now we write the polynomials in a column and add, by applying the rule for the addition of signed numbers: 2. Find the sum of 2x3-5x2+9; x2-8x3+6x -12, and x2 + 16x +24. Solution. We arrange the terms of each of the polynomials in descending powers of x (first the highest power of x, here x3, then the next highest power of x, here x2, etc.). Then we write the polynomials in a column, leaving space when there is a missing term so that like terms appear in the same column, and apply the rule for the addition of signed numbers: EXERCISES: Adding polynomials In each of the following exercises, first arrange the terms of the polynomials in order, and then arrange the polynomials in a column and find their sum: Answers : Subtraction of polynomials. Subtraction of polynomials may be done horizontally. If we are asked to subtract 4a +7b -8c + 19 from 5a +7b + 12c -15, we may first indicate the subtraction: The vertical arrangement shown below is usually more convenient. Notice that terms of the subtrahend are placed under like terms of the minuend. EXERCISES: Subtracting polynomials In each of the following problems subtract the lower polynomial from the one above it: Answers : ILLUSTRATIVE EXAMPLE: Subtraction of polynomials Subtract 7y2 - 8y3 + 45 - y from 6y3 - 14y2 - 4y. Solution. The terms of the minuend are arranged in descending powers of y. Arranging the terms of the subtrahend below the like terms of the minuend, and applying the rule for the subtraction of signed numbers: EXERCISES: Subtraction of polynomials In each of the following exercises, first arrange the terms of the polynomials in the same order; then place like terms of the first polynomial under the terms of the second polynomial and subtract: Answers : ILLUSTRATIVE EXAMPLE: Applying addition and subtraction of polynomials From the sum of y3 +6 y2 -4 and 2y2 +3 +y, subtract the sum of 8y + 6y2 +2y3 and 3y3 -7y2 -4y +2. Solution. First we find each of the two sums as explained before. EXERCISES: Adding and subtracting polynomials Answers : 1. 7a - 3c 2. - x2 + 3x - 9 3. x4 -7x3 - 8x2 + 10x - 8 4. - 2x2 + 7x 5. 15a - 9b + 3c 6. 7a - 3b + 9c 7. 36 + 2x 8. - 4b 9. - 2x - 5y 10. - 4a - 10b ILLUSTRATIVE EXAMPLES: Indicating addition or subtraction of expressions involving polynomials We learned that an expression such as 40 -(9 +2) meant that the "sum of 9 and 2" is to be subtracted from 40. In other words, the expression 40 -(9 + 2) represents the statement "40 minus the sum of 9 and 2." The illustrative examples which follow will help you express verbal statements in algebraic language. 1. Express algebraically, "x diminished by the sum of a and b." Solution. We express the sum of a and b as a+b or (a+b) Indicating that this sum is subtracted from x    x -(a+b) Answer 2. Express algebraically, "the sum of x and y increased by the sum of a and b." Solution. We express the sum of x and y as x+y or (x+y) We express the sum of a and b as a+b or (a+b) Indicating the sum of (x+y) and (a+b)                   (x+y) +(a+b) Answer 3. Express algebraically, "50 diminished by twice the sum of a and b." Solution. We express the sum of a and b as                 (a + b) We express twice the sum of a and b as                     2(a + b) Indicating that 2(a + b) is subtracted from 50            50 -2(a + b) Answer EXERCISES: Expressing verbal statements algebraically Express each of the following algebraically: 1. 25 diminished by the sum of x and y. 2. The sum of a and b diminished by 4. 3. 60 diminished by the sum of a and b. 4. The quantity x -y diminished by the quantity a-b. 5. x diminished by the sum of y and z. 6. 3x increased by the sum of a and b. 7. a diminished by the sum of 23 and x. 8. 5x diminished by twice the sum of a and b. 9. 3a increased by five times the sum of x and y. 10. The sum of x and y diminished by three times the sum of a and b. Answers : 1. 25 - (x + v) 2. (a + b) - 4 3. 60 - (a + b) 4. (x - y) - (a - b) 5. x - (y + z) 6. 3x + (a + b) 7. a - (23 + x ) 8. 5x - 2(a + b) 9. 3a + 5(x + y) 10. (x + y) - 3 (a + b)