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Cubics and Quartics

Doc Boff Further Maths AS level Inglés 15 fichas

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1What does Σ mean?The sum of....
2What does Σα represent?Sum of the roots
3What does Σαβ represent?Sum of the pairs of roots
4What does Σαβγ represent?Sum of triplets of roots
5If we expand a cubic with roots α , β and γ , what do we get?x^3 - (Σα)x^2 + (Σαβ)x - αβγ = 0
6If we expand a quartic with roots α , β , γ and δ , what do we get?x^4 - (Σα)x^3 + (Σαβ)x^2 - (Σαβγ)x + αβγδ = 0
7In a cubic, how do we work out Σα?-b/a
8In a cubic, how do we work out Σαβ?c/a
9In a cubic, how do we work out αβγ?-d/a
10In a quartic, how do we work out Σα?-b/a
11In a quartic, how do we work out Σαβ?c/a
12In a quartic, how do we work out Σαβγ?-d/a
13In a quartic, how do we work out αβγδ?e/a
14If 2x^4 - 3x^3 + x -5 = 0 and it has the roots p , q , r and s , find 1/p + 1/q + 1/r + 1/sa=2 , b=-3 , c=1 , d=-5
qrs+prs+pqs+pqr / pqrs
Σαβγ = c/a = -1/2 , αβγδ = e/a = -5/2
-1/2 / -5/2 = 1/5 , so:
1/p + 1/q + 1/r + 1/s = 1/5
15If x^3-3x^2+4x-7=0 and it has the roots α , β and γ, find α^2 + β^2 + γ^2a=1 , b=-3 , c=4 , d=7
α^2+β^2+γ^2-2(αβ+βγ+αγ)
(Σα)^2 - 2(Σαβ)
Σα = -b/a = 3 , Σαβ = c/a = 4
(3)^2 - 2(4) , 9-8 = 1 , so:
α^2 + β^2 + γ^2 = 1