Elementary Algebra and Geometry for Schools

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 HOME : Mathematics - Geometry Multiplication And Division of Polynomials The product of two monomials. We have already learned how to multiply two or more monomials. Let us review this by multiplying - 5x2y by 7x3y2. The work may be arranged either horizontally or vertically. We multiply -5 by 7 to get -35; we multiply x2 by x3 to get x5; we multiply y by y2 to get y3; and dually we indicate the product of these quantities by writing -35x55y3. Raising a monomial to a power, This is a special case of multiplication of monomials; that is, (3x)2 means (3X)(3X); (5X2)3 means (5x2)(5x2)(5x2). Thus to simplify (2X2Y3)3, we cube the numerical coefficient 2 to get 8; we cube x2 to get x6 (to cube x2 we multiply x2 . x2 . x2); we cube y3 to get y9; and finally we indicate the product of these quantities by writing 8x6y9. REVIEW EXERCISES: Multiplication of monomials. Multiply the upper monomial (the multiplicand) by the lower monomial (the multiplier).  ANSWERS : 1. 18x ; 2.35y ; 3. 9a3 ; 4. 4b4 ; 5. 27x2y ; 6. -16a4 ; 7. -30x2y ; 8. -175a3b2 ; 9. 15abxy ; 10. -69a2b2 ; 11. -12x3y2z ; 12. -32a2b4 ; 13. 42x3y5 ; 14. -48x4y3 ; 15. 81a4b6 ; 16. -63a8b9 ; 17. 15x ; 18. -6a3 ; 19. 30x2 ; 20. 8y4 ; 21. -6x3y ; 22. - 72a3b ; 23. -24m6n ; 24. 10x5y4 ; 25. 9x2 ; 26. 25a4 ; 27. 25y2 ; 28. -8x3 ; 29. -24x2y2 ; 30. 42x4 ; 31. -64a6b3 ; 32. 24a5b3 ; 33. 32x10 ; 34. 625a8b12 Multiplying a polynomial by a monomial. We have already learned to multiply a polynomial by a monomial. To multiply the polynomial 3x - 4y + 2 by the monomial 2x, we multiply each term of the polynomial by 2x and indicate the algebraic sum of these products. This work may be arranged either horizontally or vertically. We may use our knowledge of arithmetic to interpret the multiplication of a polynomial by a monomial. Thus we say that 6(a + 6 + 4) represents the area of a rectangle whose width is 6 and whose length is (a + b + 4). Let us now divide the length of the rectangle into three parts a, b, and 4. The black vertical lines at the points of division divide the rectangle into three smaller rectangles. Notice how the diagram below represents the multiplication of the polynomial a + b + 4 by the monomial 6.  As you remember, to multiply a polynomial by a monomial, multiply each tTerm of the polynomial by the monomial. REVIEW EXERCISES: Multiplying a polynomial by a monomial Multiply as indicated: Multiply the polynomial (the multiplicand) by the monomial (the multiplier). Answers : 1. 3a +3b + 3c 2. -3x3 -2x2 + x 3. -7a3 + 6a2 - 4 4. x5y3 - x3 y4 + x2y6 5. 15x3 -24x 6. -12x2 + 30x3 -6x4 7. 4a4 -20a3 -28a2 8. -12x7 + 36x6 9. -40a3 -60a2b 10. 2a4b -a3b2 + 3a2b3 11. -18x7 + 21x10 12. 36y6 + 18y9 13. -4x2 + 12x - 48 14. 6x3y -18x2y2 + 3xy3 15. 2a3b - 4a2b2 + 2ab3 16. -9a5b2 + 12a3b4 - 3a2b7 17. -a2bcd + ab2cd + abc2d -abcd2 18. 15x4y4 -15x3y5 + 5x5y6 19. -a8 + a5b3 20. -a4b + 3a3b2 -3a2b3 + ab4 Multiplying a polynomial by another polynomial. Let us first consider how to obtain the product of two binomials, each of which is numerical; for example, let us multiply 7 + 3 by 9 - 2. In this problem we can simplify each of the binomials and then find the product, thus: (7 + 3)(9 —2) (10) (7) or 70 Let us arrange the work in slightly different form, writing the first binomial as the multiplicand and the second binomial as the multiplier: 7+3 9 -2 We multiply each term of the multiplicand, 7 + 3, by 9; then we multiply each term of the multiplicand by -2; finally, we take the algebraic sum of these partial products. Thus: The first method shown above works well in arithmetic where the sum or difference of two numbers can always be completed; however, in algebra, when we are dealing with different literal numbers, their sums or their differences cannot be completed but can be indicated only. Therefore, in algebra, when we are using literal numbers, the second method shown above is used. ILLUSTRATIVE EXAMPLES: Multiplication of polynomials 1. Multiply as indicated: (3x — 4)(2x + 5) Solution. 2. Multiply 3x2 - x + 2 by 2x - 3. Check the result . Check. To check the result obtained, we let the literal quantity, here x, equal a positive number other than the number 1. Therefore, we shall let x = 2 and substitute 2 for x in each polynomial and in the product. Thus: 3. In multiplying x + 5 by x + 2, let us represent the multiplication of one polynomial by another polynomial as finding the area of a rectangle whose length and width are given. Solution : Let us divide the length of the rectangle into two segments of x and 5, and the width into two segments of x and 2. The black lines divide the rectangle into four smaller rectangles. The area of the original rectangle, of course, equals the sum of the four smaller rectangles. Notice how the diagram below represents the multiplication of x + 5 by x + 2. EXERCISES: Multiplication of polynomials Multiply the upper polynomial (the multiplicand) by the lower polynomial (the multiplier).  Multiply as indicated and check the result. Answers 1. 6x2 -11x -10. 2. 35a2 + ab -12b2 3. 6y2 -32y -70 4. 21 -26a + 8a2 5. y2 -2y + 1 6. 18x2 + 27xy -5y2 7. 14a2 -60ab + 16b2 8. 9x2 -y2 9. 4a2 + 12ab + 9b2 10. 2x3 -11x2 + 17x -6 11. 20a3 -29a2 + 56a -60 12. 3x5 + 25x4 -25x3 -63x2 13. x4 -10x3 -26x2 -6x + 5 14. x4 -2x3 + 2x2 -2x + 1 15. x4 -3x2 + 2 16. a3 + 9a2 + 23a + 12 17. 4a3 -16a2b + 17ab2 -3b3 18. 3x4 + 2x3 -19x2 -12x + 8 19. 3x2 -39x + 90 20. 2x2 + 8x -42 21. 24a2 + 34a -10 22. 4a2 -13ab + 3b2 23. a3 +1 24. 2x3 -15x2 + 31x -15 25. 3x3 -x2 +x -35. 26. 5a3 + 29a2b -11ab2 + b3 Power of a binomial. Squaring a binomial, or raising it to another power, is a special case of the multiplication of polynomials. We know that (2a)2 means (2a)(2a), and ( -3x2)3 means ( -3x2)( -3x2)( -3x2); similarly, (a -3)2means (a -3)(a -3) and (2x -y)3 means (2x -y)(2x -y)(2x -y). Thus to simplify an expression such as (3x -2y)2, we multiply 3x -2y by 3x -2y as in the preceding exercises. Thus:  To simplify (2x -y + 3)2 we multiply 2x -y + 3 by 2x -y + 3, as previously explained. Thus: Answer EXERCISES: Raising a polynomial to a power Raise each of the following to the indicated power: (2x - y)2 (3x + y)2 (x - 2y)2 (a - 4b)2 (3a - 2b)2 (6x - 4y)2 (2m + 5n)2 (5x -7y)2 (x - 9y)2 (8a + 3b)2 (a + b)3 (2x - y)3 (m + 3 n)3 (x - y + z)2 (3a + 4b - 2c)2 Answers . Special cases in the multiplication of polynomials. The special cases we shall consider are examples in which one or both polynomials are not written in descending (or ascending) powers of some letter and in which one or more terms in the sequence of exponents is missing. ILLUSTRATIVE EXAMPLE: Multiplication of polynomials Multiply x3 + 7 - 3x by 5x - 2 + 4x2 Solution. In the first polynomial the term containing x2 in the sequence of exponents is missing. First we arrange each polynomial in descending powers of x. We therefore leave a space for the missing term. Thus the first polynomial is written x3 - 3x + 7, and the second polynomial is written 4x2 + 5x - 2. Making the second polynomial the multiplier, we write: Answer EXERCISES: Multiplication of polynomials First arrange each polynomial according to the descending powers of the literal quantity and then multiply as illustrated above. Check the results obtained. 1. (2 + 3x3 — 5x)(2x — 5) 2. (2x — 4 + x2)(3x2 + 3 - 2x) 3. (5a2 + a3 - 8)(6 + 4a2 - a) 4. (2y - y2 + 4)(8 - 3y) 5. (x2 - 1 + 2x)(3 - 2x) 6. (x2 - 5 + 2x)2 7. (3x - 2 + 7x2)2 8. (8x - 1 + x3)(6 - 5x) 9. (x3 - 2 + 3x)2 10. (5x2 - x3 +2x - 2)(3x +x2 - 4)