Question

A number is called algebraic if there is a polynomial with rational coefficients for which the number is a root. For example, √2 is algebraic because it is a root of the polynomial x^2−2. The number √(2+√3+√5)is also algebraic because it is a root of a monic polynomial of degree 8, namely x^8+ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h. Find |a|+|b|+|c|+|d|+|e|+|f|+|g|+|h|.

Details and assumptions:
~A monic polynomial is a polynomial whose leading coefficient is 1.

Answer 1

If the coefficients are all rational, then the irrational roots must occur in conjugate pairs. So, the polynomial is

(x-(√2+√3+√5))
(x-(√2+√3-√5))
(x-(√2-√3+√5))
(x-(√2-√3-√5))
(x-(-√2+√3+√5))
(x-(-√2+√3-√5))
(x-(-√2-√3+√5))
(x-(-√2-√3-√5))

x^8 - 40x^6 + 352x^4 - 960x^2 + 576

. . .

Answer 2

@Steve it's not √2+√3+√5,,, it's √(2+√3+√5)...

Answer 3

[AI-generated answer — Explain Bot]

To find the values of |a|+|b|+|c|+|d|+|e|+|f|+|g|+|h| for the given polynomial x^8+ax^7+bx^6+cx^5+dx^4+ex^3+fx^2+gx+h, we need to determine the coefficients of the polynomial using the number √(2+√3+√5) as a root.

Let's go step by step:

Step 1: Simplify the given number √(2+√3+√5):
We can express √(2+√3+√5) as a sum of square roots. To do this, we rationalize the denominator by multiplying the expression by its conjugate:

√(2+√3+√5) * (√(2+√3+√5) / (√(2+√3+√5))
= (2+√3+√5) / (√(2+√3+√5))

Step 2: Consider the expression as a root of the polynomial:
Let x = √(2+√3+√5). Then the given expression can be rewritten as x^2.

Step 3: Formulate the polynomial:
Since x^2 = (2+√3+√5), we can square both sides to eliminate the square root:

x^4 = (2+√3+√5)^2
x^4 = 4 + 2√3 + 2√5 + 3 + 2√3 + 2√15 + 5 + 2√5 + 2√15
x^4 = 12 + 4√3 + 4√5 + 4√15

Step 4: Express the polynomial in terms of x:
Now we need to express x^4 in terms of x. To do this, we can substitute x^2 back in. Recall that x^2 = 2+√3+√5:

x^2 = 2+√3+√5

Multiplying both sides by x^2 yields:

x^4 = (2+√3+√5) * (2+√3+√5)
x^4 = 4 + 2√3 + 2√5 + 3 + 2√3 + 2√15 + 5 + 2√5 + 2√15
x^4 = 12 + 4√3 + 4√5 + 4√15

Step 5: Match coefficients of x^4, x^3, x^2, x, and constant term:
Next, we compare the coefficients of corresponding terms in the polynomial expression and the equation we obtained.

We have:
Coefficient of x^4: 1 (since it's a monic polynomial)
Coefficient of x^3: 0
Coefficient of x^2: 0
Coefficient of x: 0
Constant term: 12 + 4√3 + 4√5 + 4√15

Therefore:
a = 0
b = 0
c = 0
d = 0
e = 0
f = 0
g = 0
h = 12 + 4√3 + 4√5 + 4√15

Step 6: Calculate |a|+|b|+|c|+|d|+|e|+|f|+|g|+|h|:
Since a, b, c, d, e, f, g are all 0, we only need to calculate the absolute value of h:

|h| = |12 + 4√3 + 4√5 + 4√15|

Now, this involves simplifying the expression further by rationalizing the square roots. However, since the exact value of √3, √5, and √15 are not known, we cannot evaluate the expression numerically. Therefore, the exact value of |a|+|b|+|c|+|d|+|e|+|f|+|g|+|h| cannot be determined without knowing the values of √3, √5, and √15.

To conclude, the value of |a|+|b|+|c|+|d|+|e|+|f|+|g|+|h| depends on the specific values of √3, √5, and √15.