There are four positive integers that are divisors of each number in the list 36, 72, -12, 114, 96. Find the sum of these four positive integers.
Answers
Answer:
Web Homework College Algebra
Chapter 3: Polynomial Equations
3.1 Polynomials
A polynomial p(x) with coefficients in a field F is an expression of the form images/poly1.png where the coefficients images/poly2.png are in F, and n is a non-negative integer. Each of the summands is called a term of the polynomial. If images/poly3.png , then n is called the degree of the polynomial. The degree of the zero polynomial is defined to be -1. If images/poly4.png , then the coefficient images/poly5.png is called the leading coefficient of the polynomial p(x). If the leading coefficient of the p(x) is 1, the polynomial is said to be monic. The set of all polynomials with coefficients in F is denoted F[x].
A rational function of one variable is a fraction p(x)/q(x) where p and q are polynomials with coefficients in F and images/poly6.png . (Note: we mean rational functions to simply be an expression of this form; we are not thinking of it as being an actual function.) The set of all rational functions of one variable x is denoted F(x).
We can define operations on polynomials in the usual manner. The sum of two polynomials is obtained by taking the sums of the coefficients. If images/poly7.png and images/poly8.png are two polynomials, then their product is obtained by taking the sum of all the terms of the form images/poly9.png and collecting the terms with the same powers of x. This is precisely what one would do if you just expanded the product using the distributive law. One can extend in the obvious way, these operations to operations on rational functions.
Although tedious, it is straightforward to show that the rational functions of one variable with coefficients in a field F form a field.
Example 1: Let F is an ordered field. A rational function f(x) of one variable with coefficients in F can be expressed as f(x) = cp(x)/q(x) where p(x) and q(x) are monic and c is in F. We say that such a rational function is positive if and only if c is positive. One can show that this makes the field of rational functions of one variable with coefficients in F into an ordered field. It is an example of an ordered field which is NOT Archimedean.
Recall from high school algebra that one can do long division of polynomials to obtain a polynomial quotient and a polynomial remainder whose degree is smaller than that of the divisor. In summary:
Proposition 1: (Division Theorem) If p(x) and d(x) are polynomials with coefficients in a field F and images/poly10.png , then there are unique polynomials q(x) and r(x) with coefficients in F such that p(x) = q(x)d(x) + r(x) and the degree of r(x) is less than the degree of d.
Corollary 1: If f(x) is a polynomial with coefficients in a field F and a in F is a root of f(x) = 0, the f(x) = (x - a)q(x) for some polynomial q(x) with coefficients in F.
Proof: By Proposition 1, we have f(x) = (x - a)q(x) + r(x) where r(x) has degree less than 1. But then r(x) is a constant. Furthermore, 0 = f(a) = (a - a)q(a) + r(a) = r(a). So r(a) = 0. Since r is a constant, this means that r = 0.
Step-by-step explanation:
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The sum of four positive divisors of 36, 72, -12, 114, 96 = 12
Answer:
There are four positive integers that are divisors of each number in the list 36, 72, -12, 114, 96
To find:
Find the sum of four positive integer divisors of each number
Solution:
Note:
Factors of a number are its divisors.
Given numbers 36, 72, -12, 114, 96
Find HCF of 36, 72, -12, 114, 96
[ since we can't find HCF for negative number so for a while ignore negative sign of 12 ]
To find the HCF(36, 72, -12, 114, 96) write each number as product of prime number
36 = 2 × 2 × 3 × 3
72 = 2 × 2 × 2 × 3 × 3
114 = 2 × 3 × 19
96 = 2 × 2 × 2 × 2 × 2 × 3
HCF(36, 72, -12, 114, 96) = 2 × 3 = 6
The Highest Common Factor of 36, 72, -12, 114, 96 is 6
Then the factors of 6 will also be the factors 36, 72, -12, 114, 96
Factors of 6 = 1 × 2 × 3 × 6
Therefore, 4 positive divisors of 36, 72, -12, 114, 96 are 1, 2, 3, 6
Sum of the these positive divisors = 1 + 2 + 3 + 6 = 12
The sum of four positive divisors of 36, 72, -12, 114, 96 = 12
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