Need with this math problem find a polynomial function of least degree having only real coefficients, a leading coefficient of 1, and roots of 1 minus startroot 3 endroot comma 1 plus startroot 3 endroot, and 7minus i.

P(x) = x^4 -16x^3 +76x^2 -72x -100

Step-by-step explanation:

The two roots 1-√3 and 1+√3 give rise to the quadratic factor ...

... (x -(1-√3))(x -(1+√3)) = (x-1)^2 -(√3)^2 = x^2 -2x -2

The complex root 7-i has a conjugate that is also a root. These two roots give rise to the quadratic factor ...

... (x -(7 -i))(x -(7 +i)) = (x-7)^2 -(i)^2 = x^2 -14x +50

The product of these two quadratic factors is ...

... P(x) = (x^2 -2x -2)(x^2 -14x +50) = x^4 +x^3(-14 -2) +x^2(50 +28 -2) +x(-100+28) -100

... P(x) = x^4 -16x^3 +76x^2 -72x -100

Please ignore - I missed the part about coefficients must be real and did not account for the conjugate root 7+i. Sorry!

Step-by-step explanation:

Three roots, 1-sqrt3, 1+sqrt3 and 7-i, are given so the polynomial is of third degree.

To have leading coefficient of 1, simply the term (x-root) together for given roots:

P(x) = (x-(1-sqrt3)) * ((x-(1+sqrt3)) * (x-(7-i))

= (x^2-(1-sqrt3+1+sqrt3)x+(1-sqrt3)(1+sqrt3)) * (x-7+i)

= (x^2-2x+1-3) * (x-7+i)

= (x^2-2x-2) * (x-7+i)

= x^3 - (9-i)x^2 + (12-2i)x + (14-2i)

if x is real, then it can be re-arranged to real and imaginary term:

P(x) = x^3 - 9x^2 + 12x + 14 + (x^2 - 2 x - 2)i

answer: apothems is your answer

step-by-step explanation: hope u get it right

(a) common ratio in the given sequence is, 3

(b) push ups will do kendall do on the 20th week

step-by-step explanation:

geometric sequence is a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio(r)

(a)

given the sequence: 6 , 18 , 54 , 162 ,

this sequence is geometric sequence where each term is 3 times the previous term.

therefore,

common ratio(r) = 3.

(b)

a formula for nth term of geometric sequence is given by;

,   where r is the common ratio between the successive terms and is the first term,

to find how many push ups will kendall do on the 20th week.

here. n = 20th term.

in the given sequence first term

then;

simplify:

therefore, push ups will do kendall do on the 20th week