Further Algebra and Functions

Question 1

How many roots can a polynomial of order n have?

Answer 1

n roots

Question 2

How many turning points can a polynomial of order n have?

Answer 2

n - 1 turning points

Question 3

Where α and β are roots of the quadratic equation ax² + bx + c, what does α + β equal?

Answer 3

α + β = -b/a

Question 4

Where α and β are roots of the quadratic equation ax² + bx + c, what does αβ equal?

Answer 4

αβ = c/a

Question 5

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.

Answer 5

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

Question 6

Where α and β and γ are roots of the cubic equation ax³ + bx² + cx +d, what does α + β + γ equal?

Answer 6

α + β + γ = -b/a

Question 7

Where α and β and γ are roots of the cubic equation ax³ + bx² + cx +d, what does αβ + βγ + αγ equal?

Answer 7

αβ + βγ + αγ = c/a

Question 8

Where α and β and γ are roots of the cubic equation ax³ + bx² + cx +d, what does αβγ equal?

Answer 8

αβγ = -d/a

Question 9

If 1+2i and 3/4 are roots of the cubic 4x³ - 11x² + 26x - 15 = 0, what is the third root and why?

Answer 9

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.

Question 10

n
Σ (1)
r=1

Answer 10

n
Σ (1) = n
r=1

Question 11

n
Σ (r)
r=1

Answer 11

n
Σ (r) = 1/2 n (n+1)
r=1

Question 12

n
Σ (r²)
r=1

Answer 12

n
Σ (r²) = 1/6 n (n+1) (2n+1)
r=1

Question 13

n
Σ (r³)
r=1

Answer 13

n
Σ (r³) = 1/4 n² (n+1)²
r=1

Question 14

Use proof by induction to show that
n
Σ  (r) = 1/2 n (n+1)
r=1
is true for all positive integers of n.

Answer 14

1)  
For n = 1 :
n
Σ  (1) = 1
r=1
and 1/2 (1)(1+1) = 1

2) 
Assume true for n = k :
k
Σ  (r) = 1/2 k (k+1)
r=1

3)
k+1 k
Σ (r) = Σ (r) + (k+1) = 1/2 k (k+1) + (k+1)
r=1 r=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)

4)
Since true for n = 1, and if true for n = k, true for n = k + 1, must be true for all n

Question 15

Solving summations

Answer 15

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.
4) Simplify.

Question 16

Method of differences with 1/r(r+1)

Answer 16

Partial fractions
1/r(r+1) = A/r + B/r+1
1 = A(r+1) + Br
1 = Ar + A + Br
Compare coefficients:
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
= n/n+1

Question 17

Standard Maclaurin Series for sin x

Answer 17

sin x = x - x^3/3! + x^5/5! - … + (-1)^r x^(2r+1)/(2r+1)!

Question 18

Standard Maclaurin Series for cos x

Answer 18

cos x = 1 - x^2/2! + x^4/4! - … + (-1)^r x^(2r)/(2r)!

Question 19

Standard Maclaurin Series for e^x

Answer 19

e^x = 1 + x + x^2/x! + x^3/3! + … + x^r/r!

Question 20

Standard Maclaurin Series for ln (1+x)

Answer 20

ln (1+x) = x - x^2/2 + x^3/3 - … + (-1)^(r+1) x^r/r

Question 21

Standard Maclaurin Series for (1 + x)^n

Answer 21

(1 + x)^n = 1 + nx + n(n-1)x^2/2! + … + n(n-1)(n-2)…(n-r+1)x^r/r!