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8: Factoring


8: Factoring

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Scheduling Cannabis: A Preparatory Document for FDA’s 8-Factor Analysis on Cannabis

Today, over 300 million Americans live in states with some kind of law allowing for the use of medical Cannabis . However, since the passage of the Marihuana Tax Act of 1937 , federal laws and regulations surrounding the medical use of Cannabis have been based on politics rather than science. For this reason, Cannabis remains a Schedule I drug under the Controlled Substance Act (CSA). Not only does this Schedule I classification regard Cannabis as having no accepted medical use, but it also acts as a barrier towards the opportunity for research on the many potential health benefits of medical cannabis.

According to the CSA and the Food and Drug Administration’s (FDA) requirements for rescheduling a drug and recognizing its medical utility, there is more than sufficient scientific information available to begin the process of rescheduling Cannabis in the United States . For years, efforts to reschedule Cannabis based on medical and scientific information have consistently been stymied by the Drug Enforcement Administration (DEA).

In April of this year, the DEA reported to Congress that they will be issuing a response to yet another rescheduling petition by mid-year 2016. This is an ideal opportunity for the DEA to move Cannabis to a less restrictive schedule (or remove it from the CSA entirely). The CSA requires an 8-Factor analysis from the DEA for all scheduling, or rescheduling, decisions. For the purpose of showing the DEA the wealth of knowledge available on the medical benefits and low risks of medical cannabis, Americans for Safe Access (ASA) have created their own 8-Factor analysis.

The 8-Factor analysis is not a formal petition to reschedule Cannabis, rather it is a guide for addressing the government’s requirements for rescheduling and regulating Cannabis as a safe and effective medicine . The analysis presents the CSA and FDA requirements for demonstrating the medical value of legitimizing Cannabis as well as the major public health impact it can have towards the many patients it helps. Within the analysis is a critical review of the abuse, dependence, medical use, and standards being implemented to regulate medical cannabis programs in the United States.


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Hematologic Emergencies

Steven W. Salyer PA‐C , . Michael K. Shafé , in Essential Emergency Medicine , 2007

Cryoprecipitated Antihemophilic Factor (Cryoprecipitate)

Cryoprecipitate is derived from the centrifugation of 1 unit of FFP after it is slowly thawed. The unit of cryoprecipitate is then refrozen with only 10–15 ml of plasma and has a storage time of 1 year. Each unit of cryoprecipitate contains 80–120 units of factor VIII (both factor VIII:C and factor VIII:vWF) and 150–250 mg of fibrinogen and fibronectin. Cryoprecipitate was used to treat patients with hemophilia in the past, but this is no longer necessary due to the development of factor‐specific products. Cryoprecipitate is now primarily used to replace fibrinogen (below 100 mg/dl). Cryoprecipitate does not need to be type‐specific because it has such little plasma. However, hemolysis can occur after large volumes of cryoprecipitate have been given. Each unit of cryoprecipitate will raise the fibrinogen level 5–10 mg/dl. A typical dose is 10 units.


Factor VIII: structure and function in blood clotting

Factor VIII (antihemophilic factor) is the protein that is deficient or defective in patients with classical hemophilia and Von Willebrand syndrome. Factor VIII in plasma is thought to be associated in a complex with the highest molecular weight multimers of another glycoprotein, Von Willebrand protein. Highly purified human factor VIII appears to have an Mr of between 200,000 and 300,000 and to consist of several polypeptide chains. The concentration of factor VIII in plasma is around 100-200 ng/ml, equivalent to around 1 nM. The purified proteins retain one or more of the known properties of factor VIII, including the acceleration of factor IXa-mediated activation of factor X, ability to be activated by thrombin and factor Xa, inactivation by activated protein C, and by human antibodies to factor VIII. Among the known clotting factors, factors VIII and V are exceptional in not possessing enzymatic activity. Factors IXa and VIII and X appear to form a functional complex, all of which need to be present and active simultaneously for optimal activation of factor X. The mechanism by which factor VIII promotes activation of factor X by factor IXa is not known, but the major effect is to increase the rate of the reaction. Following treatment of factor VIII with thrombin, a new and smaller polypeptide Mr around 70,000 +/- 5,000 is produced. Factors IXa and Xa also have been reported to activate factor VIII. It is not known whether limited proteolytic cleavage is required absolutely for the expression of factor VIII activity or if it only increases an activity already expressed by the uncleaved protein. Factor VIII is inactivated by thrombin and by activated protein C. Thus, factor VIII can be modulated by at least four of the serine proteases in the clotting system. A major goal for future research is to increase our understanding of the role in blood clotting played by factor VIII, and to apply this information to clinical problems which result from inherited abnormalities of factor VIII.


MathHelp.com

Solve (x &ndash 3)(x &ndash 4) = 0 by factoring.

Okay, this quadratic is already factored for me. But how do I use this factorisation to solve the equation?

To solve quadratics by factoring, we use something called "the Zero-Product Property". This property says something that seems fairly obvious, but only after it's been pointed out to us namely:

Zero-Product Property : If we multiply two (or more) things together and the result is equal to zero, then we know that at least one of those things that we multiplied must also have been equal to zero. Put another way, the only way for us to get zero when we multiply two (or more) factors together is for one of the factors to have been zero.

So, if we multiply two (or more) factors and get a zero result, then we know that at least one of the factors was itself equal to zero. In particular, we can set each of the factors equal to zero, and solve the resulting equation for one solution of the original equation.

We can only draw the helpful conclusion about the factors (namely, that one of those factors must have been equal to zero, so we can set the factors equal to zero) if the product itself equals zero. If the product of factors is equal to anything non-zero, then we can not make any claim about the values of the factors.

Therefore, when solving quadratic equations by factoring, we must always have the equation in the form "(quadratic expression) equals (zero)" before we make any attempt to solve the quadratic equation by factoring.

Returning to the exercise:

The Zero Factor Principle tells me that at least one of the factors must be equal to zero. Since at least one of the factors must be zero, then I can set each of the factors equal to zero:

This gives me simple linear equations, and they're easy to solve:

And these two values are the solution they're looking for:

Note that " x = 3, 4 " means the same thing as " x = 3 or x = 4 " the only difference is the formatting. The " x = 3, 4 " format is more common.

Solve x 2 + 5x + 6 = 0 , and check.

This equation is already in the form "(quadratic) equals (zero)" but, unlike the previous example, this isn't yet factored. I MUST factor the quadratic first, because it is only when I MULTIPLY and get zero that I can say anything about the factors and solutions. I can't conclude anything about the individual terms of the unfactored quadratic (like the 5x or the 6 ), because I can add lots of stuff that totals to zero.

So the first thing I have to do is factor:

Now I can restate the original equation in terms of a product of factors, with this product being equal to zero:

Now I can solve each factor by setting each one equal to zero and solving the resulting linear equations:

These two values are the solution to the original quadratic equation. So my answer is:

I'm not done, though, because the original exercise told me to "check", which means that I need to plug my answers back into the original equation, and make sure it comes out right. In this case, I'll be plugging into the expression on the left-hand side of the original equation, and verifying that I end up with the right-hand side namely, with 0 :

When an exercise specifies that you should solve "and check", the above plug-n-chug, they're looking for you to show that you plugged your answer into the original exercise and got something that worked out right. The above, where I showed my checks, is all they're wanting. But do your work neatly!

By the way, you can use this "checking" technique to verify your answers to any "solving" exercise. So, for instance, if you're not sure of your answer to a "factor and solve" question on the next test, try plugging your answers into the original equation, and confirming that your solutions lead to true statements.

Solve x 2 &ndash 3 = 2x .

This equation is not in "(quadratic) equals (zero)" form, so I can't try to solve it yet. The first thing I need to do is get all the terms over on one side, with zero on the other side. Only then can I factor and solve:


Factor VIII

• Drug is made from human plasma and may contain infectious agents. Plasma donor screening, testing, and inactivation or removal methods reduce this risk.

Action

Promotes conversion of prothrombin to thrombin (necessary for hemostasis and blood clotting). Also replaces missing or deficient clotting factors, thereby controlling or preventing bleeding.

Availability

I.V. injection: 250, 500, 1,000, or 1,500 international units/vial in numerous preparations

Indications and dosages

➣ Spontaneous hemorrhage in patients with hemophilia A (factor VIII deficiency)

Adults and children: Dosage is highly individualized, calculated as follows: AHF required (international units) equals weight (kg) multiplied by desired factor VIII increase (% of normal) multiplied by 0.5.

To control bleeding, desired factor VIII level is 20% to 40% of normal for minor hemorrhage 30% to 60% of normal for moderate hemorrhage or 60% to 100% of normal for severe hemorrhage. To prevent spontaneous hemorrhage, desired factor VIII level is 5% of normal.

Contraindications

• Hypersensitivity to drug or to mouse, hamster, or bovine protein

Precautions

• patients receiving factor VIII inhibitors

Administration

• Before giving, verify that patient has no history of hypersensitivity to drug or to mouse, hamster, or bovine protein.

• Follow prescriber's instructions regarding hepatitis B prophylaxis before starting therapy.

• Refrigerate concentrate until ready to reconstitute drug then warm to room temperature before mixing.

• Roll bottle gently between hands until drug is well-mixed.

• Give a single dose over 5 to 10 minutes at rate of 2 to 10 ml/minute, as appropriate.

• After drug is reconstituted, don't refrigerate, shake, or store near heat.

• Don't mix with other I.V. solutions.

• Use plastic (not glass) syringe and filter.

Adverse reactions

CNS: headache lethargy fatigue dizziness jitteriness drowsiness depersonalization tingling in arms, ears, and face

CV: chest tightness, angina pectoris, tachycardia, slight hypotension, thrombosis

EENT: blurred or abnormal vision, eye disorder, otitis media, epistaxis, rhinitis, sore throat

GI: nausea, vomiting, diarrhea, constipation, stomachache, abdominal pain, gastroenteritis, anorexia,

Hematologic: forehead bruises, increased bleeding tendency, thrombocytopenia, hemolytic anemia, intravascular hemolysis, hyperfibrinogenemia

Hepatic: hepatitis B transmission Musculoskeletal: myalgia, muscle weakness, bone pain, finger pain

Respiratory: dyspnea, coughing, wheezing, bronchospasm

Skin: rash, acne, flushing, diaphoresis, urticaria

Other: taste changes, allergic reaction, fever, chills, cold feet, cold sensations, infected hematoma, stinging at injection site, anaphylaxis, human immunodeficiency virus transmission

Interactions

Drug-diagnostic tests. Bilirubin, creatine kinase: increased levels

Hemoglobin, platelets: decreased values

Patient monitoring

☞ Monitor for signs and symptoms of anaphylaxis and hemolysis.

☞ Watch for bleeding tendency and hemorrhaging.

• Check vital signs regularly.

• Monitor CBC and coagulation studies.

☞ Assess for severe headache (may indicate intracranial hemorrhage).

Patient teaching

☞ Tell patient to immediately report signs and symptoms of allergic response or bleeding tendency.

• Caution patient not to use aspirin during therapy.

• Instruct patient to contact prescriber if drug becomes less effective.

• Tell patient to report signs or symptoms of hepatitis B.

• Caution patient to avoid driving and other hazardous activities until he knows how drug affects concentration, alertness, and vision.

• Advise patient to minimize GI upset by eating small, frequent servings of food and drinking plenty of fluids.

• Notify patient that he'll undergo regular blood testing during therapy.

• As appropriate, review all other significant and life-threatening adverse reactions and interactions, especially those related to the tests mentioned above.


Rule 9: Factoring Integers

To factor an integer, simply break the integer down into a group of numbers whose product equals the original number. Factors are separated by multiplication signs. Note that the number 1 is the factor of every number. All factors of a number can be divided evenly into that number.

Example 1: Factor the number 3.

Answer: Since 3 x 1 = 3, the factors of 3 are 3 and 1.

Example 2: Factor the number 10.

Answer: Since 10 can be written as 5 x 2 x 1, the factors of 10 are 10, 5, 2, and 1. The number 10 can be divided by 10, 5, 2, and 1.

Example 3: Factor the number 18.

Answer: The number 18 can be written as 18 x 1 or 9 x 2 or 6 x 3 or 3 x 3 x 2. Since 18 can be divided by 18, 9, 6, 3, 2, and 1, then 18, 9, 6, 3, 2, and 1 are factors of 18.

Example 4: Factor the number 24.

Answer: The number 24 can be written as 24 x 1 or 12 x 2 or 8 x 3 or 4 x 6 or 2 x 2 x 2 x 3. Since 24 can be divided by 24, 12, 8, 6, 4, 3, 2, and 1, then 24, 12, 8, 6, 4, 3, 2, and 1 are factors of 24.

Example 5: Factor the number 105.

Answer: The number 105 can be written as 105 x 1 or 21 x 5 or 3 x 7 x 5 or 15 x 7 or 35 x 3. Since 105 can be divided by 105, 35, 21, 15, 7, 5, 3, and 1, then 105, 35, 21, 15, 7, 5, 3, and 1 are factors of 105.

Example 6: Factor the number 1200 completely.

Answer: This instruction means to factor 1200 into a set of prime factors (factors that cannot again be factored). The number 1200 can be written as 1200 x 1 or 100 x 12. Note the 100 can again be factored to 10 x 10 and the 12 can be factored to 6 x 2. So now you have 1200 = 100 x 12 = 10 x 10 x 6 x 2. This factored set can again be factored to (2 x 5) x (2 x 5) x (2 x 3) x 2 x 1. The number 1200 is factored completely as 5 x 5 x 3 x 2 x 2 x 2 x 2 x 1.


Usual Pediatric Dose for Hemophilia A

ADVATE(R), HEXILATE FS(R), Kogenate FS(R):
Dose (IU) = Weight (kg) x Desired factor VIII rise (% normal or IU/dL) x 0.5
Expected factor VIII rise (% of normal): 2 x IU administered / weight (kg)

Control of Bleeding Episodes:
Minor bleeding (early hemarthrosis, minor muscle or oral bleeds): 10 to 20 IU/kg intravenously until bleeding is resolved repeat for evidence of further bleeding
-Factor VIII level required = 20 to 40% of normal

Moderate bleeding (hemorrhage into muscles/oral cavity, hemarthrosis, known trauma): 15 to 30 IU/kg intravenously every 12 to 24 hours until bleeding is resolved.
-Factor VIII level required = 30 to 60% of normal

Major bleeding (gastrointestinal, intracranial, intraabdominal, intrathoracic, central nervous system, retropharyngeal, retroperitoneal, or iliopsoas sheath bleeds, fractures, head trauma):
-Initial dose: 40 to 50 IU/kg intravenously
-Repeat dose: 20 to 25 IU/kg intravenously every 8 to 12 hours until bleeding is resolved.
-Factor VIII level required = 80 to 100% of normal

Surgery:
Minor surgery (including tooth extraction): 15 to 30 IU/kg intravenously every 12 to 24 hours until bleeding is resolved.
-Factor VIII level required = 30 to 60% of normal
Major surgery (e.g. tonsillectomy, inguinal herniotomy, synovectomy, craniotomy, osteosynthesis, trauma, joint replacement):
-Prior to surgery: 50 IU/kg intravenously to achieve 100% factor VIII activity continue dosing every 6 to 12 hours to keep factor VIII activity in desired range until healing is complete

Routine Prophylaxis: 25 IU/kg every other day

AFSTYLA(R), HEMOFIL M(R), NOVOEIGHT(R), NUWIQ(R), RECOMBINATE(R), XYNTHA(R):
Dose (IU) = Weight (kg) x Desired factor VIII rise (% normal or IU/dL) x 0.5
Expected factor VIII rise (% of normal): 2 x IU administered / weight (kg)

Control and Prevention of Bleeding Episodes:
Minor bleeding (superficial muscle, soft-tissue, or oral): dose to post-infusion factor VIII level of 20 to 40% every 12 to 24 hours for at least one day, until bleeding is resolved

Moderate to major bleeding (hemorrhage into muscles/oral cavity, hemarthrosis, known trauma): dose to post-infusion factor VIII level of 30 to 60% every 12 to 24 hours for 3 to 4 days or longer until bleeding is resolved.

Life-threatening bleeding (hemorrhage into muscles/oral cavity, hemarthrosis, known trauma): dose to post-infusion factor VIII level of 60 to 100% every 8 to 24 hours until bleeding is resolved.

Surgery:
Minor surgery (including tooth extraction): dose to post-infusion factor VIII level of 30 to 60% every 24 hours, for at least 1 days until healing is achieved.
Major surgery (intracranial, intraabdominal, joint replacement): dose to post-infusion factor VIII level of 80 to 100% every 8 to 24 hours until wound healing, the continue therapy at least 7 more days maintaining factor VIII level of 30 to 60%.

Routine Prophylaxis:
12 years and older: 30 to 40 IU/kg every other day
Under 12 years: 30 to 50 IU/kg every other day or three times per week


ELOCTATE(R):
Dose (IU) = Weight (kg) x Desired factor VIII rise (% normal or IU/dL) x 0.5
Expected factor VIII rise (% of normal): 2 x IU administered / weight (kg)

Control of Bleeding Episodes:
Minor to moderate bleeding (joint, superficial muscle without neurovascular compromise except iliopsoas, deep laceration, renal, superficial soft tissue, mucous membranes): 20 to 30 IU/kg intravenously every 24 to 48 hours until bleeding is resolved.
-Factor VIII level required = 40 to 60% of normal

Major bleeding (life/limb threatening hemorrhage, iliopsoas and deep muscle with neurovascular injury, retroperitoneal, intracranial, gastrointestinal): 40 to 50 IU/kg intravenously every 12 to 24 hours until bleeding is resolved (approximately 7 to 10 days)
-Under 6 years: Repeat dose every 8 to 24 hours until bleeding is resolved.
-Factor VIII level required = 80 to 100% of normal

Surgery:
Minor surgery (including tooth extraction): 25 to 40 IU/kg intravenously repeat every 24 hours if needed for a duration of at least 1 day until healing is achieved.
Major surgery (intracranial, intraabdominal, joint replacement):
-Prior to surgery: 40 to 60 IU/kg intravenously once prior to surgery
-After surgery: 40 to 50 IU/kg intravenously 8 to 24 hours after surgery (6 to 24 hours in patients under 6 years), then every 24 for until adequate wound healing, then continue at least 7 days while maintaining factor VIII within target range.
-Factor VIII level required prior to and after surgery= 80 to 120% of normal

Routine Prophylaxis:
-6 years and older, initial dose: 50 IU/kg every 4 days
-Under 6 years, initial dose: 50 IU/kg twice a week
-Maintenance dose (all ages): 25 to 65 IU/kg every 3 to 5 days
-More frequent or higher doses (up to 80 IU/kg) may be required.
-Adjust frequency based on bleeding episodes.


ESPEROCT(R):
Dose (IU) = Weight (kg) x Desired factor VIII rise (% normal or IU/dL) x 0.5

Control of Bleeding Episodes:
Minor bleeding (early hemarthrosis, minor muscle or oral bleeds):
-12 years and older: 40 IU/kg intravenously one dose should be sufficient
-Under 12 years: 65 IU/kg intravenously one dose should be sufficient

Moderate bleeding (more extensive hemarthrosis, muscle bleeding, hematoma):
-12 years and older: 40 IU/kg intravenously may give one additional dose after 24 hours
-Under 12 years: 65 IU/kg intravenously may give one additional dose after 24 hours

Major bleeding (life or limb threatening, gastrointestinal, intracranial, intraabdominal, intrathoracic, fractures):
-12 years and older: 40 IU/kg intravenously may repeat approximately every 24 hours
-Under 12 years: 65 IU/kg intravenously may repeat approximately every 24 hours

Surgery:
Minor surgery (including tooth extraction): 50 IU/kg intravenously may repeat after 24 hours if needed
Major surgery (intracranial, intraabdominal, intrathoracic, joint replacement):
-12 years and older: 50 IU/kg intravenously may repeat approximately every 24 hours for the first week, then approximately every 48 hours until wound healing has occurred.
-Under 12 years: 65 IU/kg intravenously may repeat approximately every 24 hours for the first week, then approximately every 48 hours until wound healing has occurred.

Routine Prophylaxis:
-12 years and older, initial dose: 50 IU/kg every 4 days
-Under 12 years: 65 IU/kg twice a week
-Adjust frequency based on bleeding episodes.


KOATE(R):
Dose (IU) = Weight (kg) x Desired factor VIII rise (% normal or IU/dL) x 0.5

Control and Prevention of Bleeding Episodes:
Minor bleeding (large bruises, significant cuts/scrapes, uncomplicated joint hemorrhage): 15 IU/kg intravenously every 12 hours (twice daily) until hemorrhage stops and healing achieved (1 to 2 days)
-Factor VIII level required = 30% of normal

Moderate bleeding (nose/mouth/gum bleeds, dental extractions, hematuria): 25 IU/kg intravenously every 12 hours (twice daily) until healing achieved (average of 2 to 7 days)
-Factor VIII level required = 50% of normal

Major bleeding (joint hemorrhage, muscle hemorrhage, major trauma, hematuria, intracranial/intraperitoneal bleeding):
Initial dose: 40 to 50 IU/kg intravenously every 12 hours (twice daily)
Maintenance dose: 25 until healing achieved (up to 10 days)
-Intracranial hemorrhage may require prophylaxis for up to 6 months.
-Factor VIII level required = 80 to 100% of normal

Surgery:
Prior to surgery: 40 to 50 IU/kg intravenously once prior to surgery
After surgery: 40 to 50 IU/kg intravenously for 7 to 10 days or until healing achieved
-Factor VIII level required prior to surgery= 80 to 100% of normal
-Factor VIII level required after surgery= 60 to 100% of normal


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