Flashcards in Further Algebra and Functions Deck (21)
How many roots can a polynomial of order n have?
How many turning points can a polynomial of order n have?
n - 1 turning points
Where α and β are roots of the quadratic equation ax² + bx + c, what does α + β equal?
α + β = -b/a
Where α and β are roots of the quadratic equation ax² + bx + c, what does αβ equal?
αβ = c/a
The roots of 2z² + 3z + 5 = 0 are α and β.
i) Find values for α + β and αβ.
ii) Form a quadratic equation with roots 2α and 2β.
iii) Do this with a different method.
i) α + β = -b/a = -3/2
αβ = c/a = 5/2
ii) 2α + 2β = 2(α + β) = 2(-3/2) = -3
2α * 2β = 4αβ = 4(5/2) = 10
Let a be 1:
a = 1
b = 3
c = 10
z² + 3z + 10 = 0
iii) x = 2α
α = x/2
2(x/2)² + 3(x/2) + 5 = 0
2x²/4 + 3x/2 + 5 =0
x² + 3x + 10 = 0
Where α and β and γ are roots of the cubic equation ax³ + bx² + cx +d, what does α + β + γ equal?
α + β + γ = -b/a
Where α and β and γ are roots of the cubic equation ax³ + bx² + cx +d, what does αβ + βγ + αγ equal?
αβ + βγ + αγ = c/a
Where α and β and γ are roots of the cubic equation ax³ + bx² + cx +d, what does αβγ equal?
αβγ = -d/a
If 1+2i and 3/4 are roots of the cubic 4x³ - 11x² + 26x - 15 = 0, what is the third root and why?
The third root is 1-2i because the complex conjugate of a root is also a root, unless one of the coefficients of the equation is complex.
Σ (1) = n
Σ (r) = 1/2 n (n+1)
Σ (r²) = 1/6 n (n+1) (2n+1)
Σ (r³) = 1/4 n² (n+1)²
Use proof by induction to show that
Σ (r) = 1/2 n (n+1)
is true for all positive integers of n.
For n = 1 :
Σ (1) = 1
and 1/2 (1)(1+1) = 1
Assume true for n = k :
Σ (r) = 1/2 k (k+1)
Σ (r) = Σ (r) + (k+1) = 1/2 k (k+1) + (k+1)
1/2 k (k+1) + (k+1) = 1/2 (k (k+1) + 2(k+1)) = 1/2 (k² + 3k +2) = 1/2 (k+1)((k+1)+1)
Since true for n = 1, and if true for n = k, true for n = k + 1, must be true for all n
1) Expand brackets and rearrange to get the expression in the form of a polynomial.
2) Split up the summation into sums of terms; e.g. 2Σ(r²).
3) Substitute the formulas.
Method of differences with 1/r(r+1)
1/r(r+1) = A/r + B/r+1
1 = A(r+1) + Br
1 = Ar + A + Br
A = 1 B = -1
1/r(r+1) = 1/r - 1/r+1
Method of differences
Σ 1/r(r+1) = Σ 1/r - 1/r+1
= 1/1 - 1/2
+ 1/2 - 1/3
+ 1/3 - 1/4 + ...
+ 1/n-1 - 1/n
+ 1/n - 1/n+1
= 1 - 1/n+1
Standard Maclaurin Series for sin x
sin x = x - x^3/3! + x^5/5! - ... + (-1)^r x^(2r+1)/(2r+1)!
Standard Maclaurin Series for cos x
cos x = 1 - x^2/2! + x^4/4! - ... + (-1)^r x^(2r)/(2r)!
Standard Maclaurin Series for e^x
e^x = 1 + x + x^2/x! + x^3/3! + ... + x^r/r!
Standard Maclaurin Series for ln (1+x)
ln (1+x) = x - x^2/2 + x^3/3 - ... + (-1)^(r+1) x^r/r