University:
University of Michigan
Course:
MATH 115 | Calculus I
Academic year:

2024
Views:

284

Pages:

2
Author:

Luciano Kirby

Question 1 When to stop dividing in the long division of polynomials. Answer The four stages used to divide whole numbers-(divide, multiply, subtract, bring down the next term)-form the repeated process for the polynomial long division when a divisor has more than one term or if the divisor is a polynomial having more than one term. The stages for dividing by a polynomial (or a lengthy division of polynomials) are listed below: Sort the dividend and divisor's terms in decreasing powers of any variable. Divide the dividend's first term by the divisor's first term. The first term of the quotient is the outcome. Multiplying each phrase in the divisor by the quotient's first term Similar phrases should be put up beneath the dividend and the resulting product. Subtract the dividend from the product. To create a new dividend, lower the next term from the old dividend and write it next to the remaining terms. Repeat this procedure using this new expression as the dividend until the remaining can no longer be split. When the degree of the remainder-the exponent with the largest value on a variable in the remainder-is lower than the degree of the divisor, this will happen. divisor = x² + 1, Remainder = x + 5 In this case, the residual has a degree of "one," which is lower than the divisor's degree of "two." At this point, division should cease since the remaining portion cannot be split further. Question 2 Synthetic division is a process for dividing a polynomial by x - c. The coefficient of x in the divisor is 1. How might synthetic division be used if you are dividing by 2x - 4? Answer Step 1 Synthetic division is a method of polynomial division of a cubic polynomial in linear form. Synthetic division is generally used however, not for dividing out factors but for finding the roots (or zeroes) of polynomials. Assume, y = x² + 5x + 6, be a polynomial then we can factor the polynomial as y=(x+3)(x + 2). This polynomial can be solved by using a simple basic algebraic formula x = -b±√b²-4ac / 2a Step 2 The coefficient of x in the divisor is 1, as already been said Synthetic division is generally used however, not for dividing out factors but for finding the roots (or zeroes) of polynomials. Hence, the coefficient of x is of less concern and we cannot divide any polynomial with another polynomial unless otherwise it is factored. Synthetic division cannot be used if the dividing polynomial coefficient has some constant values. Question 3 Synthetic division is a shortcut process for polynomial division. Answer Approach: The long division method and synthetic division method are used to divide polynomials. In synthetic division, we divide the term's coefficients. This procedure is a shortcut for a polynomial division. Calculation: A polynomial can be divided using long division and the synthetic division method. In synthetic division we take the coefficients of the terms and complete the division. The divisor must be in the form of x - k. Compared to the long-division method, it is simpler to solve. The synthetic division is a short cut process for a polynomial division as it is simple and easy to solve. Thus, synthetic division is a shortcut for a polynomial division. Therefore, the given statement is a true statement. Final statement: The answer for the statement is true. 8.8s