56 CHAPTER R Review of Basic Concepts The quotient of two polynomials, where the divisor has more than one term, can be found with an algorithm (that is, a step-by-step procedure) for long division similar to that used for dividing whole numbers. Both polynomials must be written in descending order to use this algorithm. EXAMPLE 7 Dividing a Polynomial by a Monomial Divide 16a5 - 12a4 + 8a2 by 4a3. SOLUTION 16a5 - 12a4 + 8a2 4a3 = 16a5 4a3 - 12a4 4a3 + 8a2 4a3 Divide each term by 4a3. = 4a2 - 3a + 2 a 8a2 4a3 = 8 4a2-3 = 2a-1 = 2 a CHECK 4a3a4a2 - 3a + 2 ab = 16a5 - 12a4 + 8a2 ✓ Divisor * Quotient = Dividend S Now Try Exercise 79. This becomes 2 a, not 2a. Division Dividing a polynomial by a monomial uses the rules for exponents. EXAMPLE 8 Dividing a Polynomial by a Monomial Divide -180x4y10 + 90xy4 - 100y by -30x2y2. SOLUTION -180x4y10 + 90xy4 - 100y -30x2y2 = -180x4y10 -30x2y2 + 90xy4 -30x2y2 - 100y -30x2y2 Divide each term by -30x2y2. = 6x2y8 - 3y2 x + 10 3x2y S Now Try Exercise 85. EXAMPLE 9 Dividing Polynomials Divide 4m3 - 8m2 + 5m+ 6 by 2m- 1. SOLUTION 4m3 divided by 2m is 2m2. -6m2 divided by 2m is -3m. 2m divided by 2m is 1. 2m2 - 3m+ 1 2m- 1)4m3 - 8m2 + 5m+ 6 4m3 - 2m2 2m212m- 12 = 4m3 - 2m2 -6m2 + 5m Subtract. Bring down the next term. -6m2 + 3m -3m12m- 12 = -6m2 + 3m 2m+ 6 Subtract. Bring down the next term. 2m- 1 112m- 12 = 2m- 1 7 Subtract. The remainder is 7. Thus, 4m3 - 8m2 + 5m+ 6 2m- 1 = 2m2 - 3m+ 1 + 7 2m- 1 . S Now Try Exercise 97. To subtract, add the opposite. Remember to add remainder divisor .
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