EXERCISES Perform the following divisions and check: 1. (a2 2ab+b2) ÷ (a — b) 8. (2t+u) (-14r2 + 25rs 6s2) ÷ (— 7r + 2s) 9. (-4ab3a2 + b2) ÷ (3a — b) 10. (xy + 6x2 11. (a3 + b3) ÷ (a + b) 15y2) ÷ (3x+5y) 15. (a5 + b3) ÷ (a + b) 24. 19. (8ab + b2 + 7a2) ÷ (a + b) 20. (3xy+9x2 + y2) ÷ (3x+2y) 21. (27a3 (27a35a2 + 36a - 8) ÷ (3a - 2) 22. (ab 23. (a2+2ab+b2 - c2) ÷ (a + b c) Ex A rectangle 13 ft. by 17 ft. has each dimension increased ƒ feet. press its length and width and find its area after the increase. 25. Find the area of a circle whose rádius is 4 ft. If the radius is increased to (4+f) feet, what is then the area? What is the increase in area? 26. The area of a rectangle is a2 + 11a + 28 sq. in. Find the width if the length is (a + 7) inches. Hold a number contest on division of polynomials. 68. Equations Containing Signs of Grouping.-Equations containing signs of grouping often arise. In all such equations it is first necessary to remove the signs of grouping. Check the root by substituting it in (1). 21. 15g (8g + 5) = 30 (12r+2) 6 = 20 = 13 7. = 10 8. = 45 9. 10. 14B = 2a) 11 (2k+3)= 37 (3t + 4) = 61 10A + 2(3A + 5) = 90 (− 5a + 2) — (3a 13. 14. 13 Find a number n such that if 13 less than the number be multiplied by 3 and 5 times the number be added to this, the result is 17. Set up the equation, solve, and verify. 22. Four times the sum of a number and 5 equals 32. Find the number. 23. If 7 be subtracted from a number and this remainder be multiplied by 3, the result will be 6. Find the number. 69. Equations.--Equations may contain signs of grouping, as (m3)(m + 5). The signs of grouping are then removed by carrying out the indicated operation. 7. (m2 - 6m+9) ÷ (m − 3) = 4 (g+5)(g5) + 2g g2 — 7 8. = 10. If one side of a square is increased 2 ft. and the other 3 ft., the area of the rectangle formed equals the area of the square and 36 sq. ft. more. Find the side of the square. 11. Express the length and width of a rectangle that is 8 ft. longer than it is wide. A new rectangle formed by taking 4 ft. from the length and adding 2 ft. to the width has an area equal to the original rectangle. Find the width. III FACTORS AND SPECIAL PRODUCTS 70. Factoring.-Finding those numbers which multiplied together produce a given number is called factoring. Thus, we factor 15 when we note that it is composed of 5 X 3. The 3 and 5 are the factors of 15. Some literal numbers can be factored by finding a monomial factor of each term. Thus, 10a2b3 + 15a3c has coefficients divisible by 5 and the literal part divisible by a2 so that 10a2b3+15a3c5a2(2b+3ac). The factors of 10a2b3 + 15a3c are then 5a2 and 2b3 + 3ac. Factoring is really the process of undoing multiplication. 71. Prime and Composite Numbers.-Numbers which have no factors are called prime numbers. Such are 11,17, 31, ab, 3c - 7m, etc. 7m, etc. Name some others. Numbers which have factors are called composite numbers. Such are 12, 35, 5a2c, 3m2 6mn, etc. Name some = 4, the square root 72. Square Roots.-The square root of a number is one of its two equal factors. Since 2 × 2 of 4 is 2. But since (-2) × (− 2) of 4 is also 2. That is, √4 = = 4, 2, or the square root 2. Hence, a number has two square roots with opposite signs. 73. Quadratic Equations.-The area of a circle is found by replacing R with its numerical value in, To find the radius of the circle whose area is Equations like (2), (3), (4) are called quadratic equations. EXERCISES 1. Give the square roots of 81; 49; 1; 625; 144; 36. Find the two roots in each of the following: 15. Make up three examples like the above for the other members of your class to solve. |