Question

Determine the least degree monic polynomial p of x with coefficients in Q such that (square root of 2) is root of p of x and 2+i is multiple root of p of x

132

likes
661 views

Answer to a math question Determine the least degree monic polynomial p of x with coefficients in Q such that (square root of 2) is root of p of x and 2+i is multiple root of p of x

Expert avatar
Jayne
4.4
102 Answers
Para encontrarmos o polinômio mônico de menor grau com coeficientes em \mathbb{Q} sabendo que \sqrt{2} é raiz de p(x) e 2+i é raiz múltipla de p(x) , podemos utilizar essas informações para construir as raízes de p(x) .

1. Como \sqrt{2} é raiz de p(x) , então x-\sqrt{2} é um fator de p(x) .
2. Como 2+i é raiz de p(x) , então x-(2+i) é um fator de p(x) . Como é raiz múltipla, temos outro fator x-(2-i) .
Portanto, temos até agora: (x-\sqrt{2})(x-(2+i))(x-(2-i)) .

Depois, multiplicamos os fatores para obter o polinômio:
p(x)=(x-\sqrt{2})(x-(2+i))(x-(2-i))
p(x)=(x-\sqrt{2})(x-2-i)(x-2+i)
p(x)=(x-\sqrt{2})(x^2-4x+4i)
p(x)=x^3-4x^2+4ix-\sqrt{2}x^2+4\sqrt{2}x-4\sqrt{2}i
p(x)=x^3-(4+\sqrt{2})x^2+(4\sqrt{2}+4i)x-4\sqrt{2}i

Portanto, o polinômio mônico p(x) de menor grau com coeficientes em \mathbb{Q} tal que \sqrt{2} é raiz de p(x) e 2+i é raiz múltipla de p(x) é:
\boxed{p(x) = x^3-(4+\sqrt{2})x^2+(4\sqrt{2}+4i)x-4\sqrt{2}i}

Frequently asked questions (FAQs)
What is the area of a triangle with base length 5 and height 8?
+
What is the equation of a line passing through (2,4) and (6,10)?
+
Mean Quiz: What is the average of 7, 13, 15, 18?
+
New questions in Mathematics
If we have the sequence: 3, 6, 12, 24 Please determine the 14th term.
find the value of the tangent if it is known that the cos@= 1 2 and the sine is negative. must perform procedures.
Calculate the 6th term of PA whose 1st term is 6.5 and the ratio 5
I) Find the directional derivative of 𝑓(𝑥, 𝑦) = 𝑥 sin 𝑦 at (1,0) in the direction of the unit vector that make an angle of 𝜋/4 with positive 𝑥-axis.
what is the annual rate on ​$525 at 0.046​% per day for 3 months?
Find all real numbers x that satisfy the equation \sqrt{x^2-2}=\sqrt{3-x}
What’s the slope of a tangent line at x=1 for f(x)=x2. We can find the slopes of a sequence of secant lines that get closer and closer to the tangent line. What we are working towards is the process of finding a “limit” which is a foundational topic of calculus.
reduce the expression (7.5x 12)÷0.3
89, ÷ 10
How to do 15 x 3304
When Sara was 15 years old, an uncle left her as inheritanceà a sum of 10,000 euros which he invested in a bank that applies the interest rate of 2,5% annual. Today Sara is 18 years and wants to buy a'car, how much she can ò withdraw from the bank?
Associate each 2nd degree equation with its respective roots. A) x2+6x+8=0 B)x2-5x-6=0
The average undergraduate cost per tuition, fees, room, and board for all institutions last year was $26,025. A random sample of 40 institutions of higher learning this year indicated that the mean tuition, fees, room, and board for the sample was $27,690, and the population standard deviation is $5492. At the 0.05 level of significance, is there sufficient evidence that the cost has increased? (Remember to follow the steps in hypothesis testing)
Consider the function f(x)=1/2(x+1)^2-3. Use the preceding/following interval method to estimate the instantaneous rate of change at 𝑥 = 1.
Find I (Intrest) using simple interest formula of 17700 @ 15% for 4 years
8. Measurement Jillian measured the distance around a small fish pond to be 27 yards. What would be a good estimate of the distance across the pond: 14 yards, 9 yards, or 7 yards? Explain how you decided.
g(x)=3(x+8). What is the value of g(12)
A person travels by car from one city to another with different constant speeds between pairs of cities. She drives for 55.0 min at 100.0 km/h, 14.0 min at 65.0 km/h, and 45.0 min at 60.0 km/h and spends 20.0 min eating lunch and buying gas. (a) Determine the average speed for the trip.
Find the number of liters of water needed to reduce 9 liters of lotion. shave containing 50% alcohol to a lotion containing 30% alcohol.
Construct a set of six pieces of data with​ mean, median, and midrange of 67 and where no two pieces of data are the same.