9. If f(x) # 0f and f(x) is relatively prime to Or, what can be said about f(x)?
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
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(g) 2x* + 5x³ – 5x - 2 and 2x – 3x² – 2x in Q[x].
6. Express each of the gcd's in Exercise 5 as a linear combination of the two
polynomials.
B. 7. Let f(x) E F[x] and assume that f(x)|g(x) for every nonconstant g(x)E F[x]. Show
that f(x) is a constant polynomial. [Hint: f(x) must divide both x + 1 and x.]
8. Let f(x), g(x)E F[x], not both zero, and let d(x) be their gcd. If h(x) is a
common divisor of f(x) and g(x) of highest possible degree, then prove that
h(x) = cd(x) for some nonzero cEF.
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9. If f(x) # OF and f(x) is relatively prime to Or, what can be said about f(x)?
10. Find the gcd of x + a + b and x3 – 3abx + a³ + b³ in Q[x].
11. Fill in the details of the proof of Theorem 4.8.
12. Prove Corollary 4.9.
13. Prove Theorem 4.10.
Coutseome1.
14. Let f(x), g(x), h(x)E F[x], with f(x) and g(x) relatively prime. If f(x)|h(x) and
g(x) |h(x), prove that f(x)g(x)|h(x).
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