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1.BIOL11 Exam 2 Study Guide Flashcards – Quizlet2.10 quiz Flashcards – Quizlet3.Solved QUESTION 31 A neutral mutation changes the codon CUUThe information shared above about the question what dna sequence is complementary to cggtgaactgta, certainly helped you get the answer you wanted, please share this article to everyone. so that everyone can know this useful information. Wish you a good day! Top EN -
12 The remainder theorem The factor theorem IN ARITHMETIC we write, for example,  or, Equivalently, 47 = 9· 5 + 2 5 is called the divisor, 47 is the dividend, 9 is the quotient, and 2 is the remainder.
Or, Dividend = Quotient· Divisor + Remainder. In algebra, if we divide a polynomial P(x) by a polynomial D(x) (where the degree of D is less than the degree of P), we would find P(x) = Q(x)· D(x) + R(x). P(x) is the dividend, Q(x) is the quotient, and R(x) is the remainder. For example, if, by long division, we divided x3 − 5x2 + 3 x − 7 by x − 2, we would find
Or, x3 − 5x2 + 3x − 7 is the dividend, x2 − 3x −3 is the quotient, and −13 is the remainder. Here is how to do this problem by synthetic division. First, to use synthetic division, the divisor must be of the first degree and must have the form x − a. In this example, the divisor is x − 2, with a = 2. Here again is the problem: x − 2 Proceed as follows: 1. Write the coefficients of the dividend: 1 − 5 + 3 − 7 2. Put a, in this case 2, in a box to the right, leave a space, and draw a 3. Bring down the leading coefficient (1), multiply it with a (2), and 4. Add: 5. Repeat the process. −3· 2 = −6. And so on, until all the coefficients The first three numbers, 1 − 3 − 3, are the coefficients of the quotient, and the final number, −13, is the remainder. We have x3 − 5x2 + 3 x − 7 = (x2 − 3x −3)(x − 2) − 13. Example 1. Use synthetic division to divide 2x5 + 3x4 + 25x2 − 1 by x + 3. Solution. There are a couple of points here. First, we must account for all six coefficients of the general form. 2 + 3 + 0 + 25 + 0 − 1 The coefficient of x3 is 0, as is the coefficient of x. Next, the divisor is x + 3. But the divisor must have the form x − a. x + 3 = x − (−3). Therefore, a = −3. Here is the synthetic division: This tells us
Or,
Note: The degree of the quotient is one less than the degree of the dividend. And the degree of the remainder is less than the degree of the divisor, x + 3, which in this case is 1. The remainder therefore is of degree 0, which is a number. In general, if we divide a polynomial of degree n by a polynomial of degree 1, then the degree of the quotient will be n − 1. And the remainder will be a number. Problem 1. Use synthetic division to divide x3 − 8x2 + x + 2 by x − 7. Write your answer in the form P(x) = Q(x)· D(x) + R. To see the answer, pass your mouse over the colored area. x3 − 8x2 + x + 2 = (x2 − x − 6)(x − 7) − 40 The remainder theorem The value of a polynomial P(x) at x = a, P(a), is equal to the remainder upon dividing P(x) That is, when P(x) = Q(x)(x − a) + R, where Q(x) is the quotient and R is the remainder, then P(a) = R. For,
Example 2. Let f(x) = x3 − 3x2 − 13x + 15. We will use synthetic division to divide f(x) by x + 4. Now, what does the remainder theorem tell us? The value of f(x) at x = −4, is equal to the remainder: f(−4) = −45. Now let us divide f(x) by x − 5: What does the remainder theorem tell us here? f(5) = 0. But this means that 5 is a root of f(x) Moreover, since the remainder is 0 -- there is no remainder -- then (x − 5) is a factor of f(x). The synthetic division shows: x3 − 3x2 − 13x + 15 = (x2 + 2x − 3)(x − 5) This illustrates the Factor Theorem: The Factor Theorem. x − r is a factor of a polynomial P(x) if and only if r is a root of P(x). Problem 2. Let f(x) = x3 − 5x2 − 4x + 7. Use synthetic division to divide f(x) by x − 7. Therefore, according to the remainder theorem, f(7) = 77. Since the remainder is not 0 -- f(7) Problem 3. Let g(x) = 3x 4 + 17x3 + 16x 2 − 10x + 4. Use synthetic division to divide g(x) by x + 2. According to the remainder theorem, g(−2) = 0. Therefore, what do you conclude about −2? −2 is a root of g(x). What do you conclude about (x + 2)? (x + 2) is a factor of g(x). Problem 4. Use synthetic division to divide x3 + 125 by x + 5. x3 + 125 = (x2 − 5x + 25)(x + 5) Next Topic: Roots of polynomials Table of Contents | Home Please make a donation to keep TheMathPage online. Copyright © 2021 Lawrence Spector Questions or comments? E-mail: [email protected] What divisor represents the synthetic division below?The divisor represented by the synthetic division below is x + 5.
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