Chapter 2 - Algebra and Series

Question 1

For a quadratic equations, what are the rules for finding the values of the coefficients of an equation where x has roots α and β?

Answer 1

Question 2

For a cubic equations, what are the rules for finding the values of the coefficients of an equation where x has roots α β and γ?

Answer 2

For an equation ax³ + bx² + cx + d = 0, where x has roots α β γ:

α + β + γ = -b/a
(α x β) + (β x γ) + (γ x α) = c/a
α x β x γ = -d/a

Question 3

For quartic equations, what are the rules for finding the values of the coefficients of an equation where x has roots α β γ and 𝛿?

Answer 3

For an equation ax⁴ + bx³ + cx² + dx + e = 0, where x has roots α β γ 𝛿:

α + β + γ + 𝛿 = -b/a
(α x β) + (α x γ) + (α x 𝛿) + (β x γ) + (β x 𝛿) + (γ x 𝛿) = c/a
(α x β x γ) + (β x γ x 𝛿) + (γ x 𝛿 x α) + (𝛿 x α x β) = -d/a
α x β x γ x 𝛿= e/a

Question 4

How would you describe an equation if the roots are transformed in a linear way, so that y = mx + c, or if an equation is a reciprocal?

Answer 4

If the roots are transformed in a linear way, so that y = mx+c, then you transform the equation by substituting x = (y-c)/m

If the new roots are reciprocals, y = 1/x, then you transform the equation by substituting x = 1/y.

Question 5

How do you represent the sum of R, from 1 to n?

n
∑r = ?
1

Answer 5

The formula for the sum of R from 1 to n is:

n
∑r = n(n+1) /2
1

Question 6

How do you represent the sum of R², from 1 to n?

n
∑r² = ?
1

Answer 6

The formula for the sum of R from 1 to n is:

n
∑r² = n(n+1)(2n+1)/6
1

Question 7

How do you represent the sum of R³, from 1 to n?

n
∑r³ = ?
1

Answer 7

The formula for the sum of R from 1 to n is:

n
∑r³ = (n²)(n+1)² /4
1

Question 8

What are the steps for Proof by Induction?

Answer 8

Proof by induction is method that is used to prove a mathematical statement for all natural numbers (positive integers).

The three key steps for a proof by induction are:

1. Prove the statement is true for n=1.
2. Assume the statement is true for n=k, and use this to prove the statement is true for n = k+1.
3. Write a conclusion.

Question 9

What is the definition of a Maclaurin series?

Answer 9

The definition of a Maclaurin series is:

f(x) = f(0) + xf’(0) + x²/2!(f’‘(0)) …. xʳ/r!(f ʳ(0))

Question 10

What are the five Maclaurin Series functions?

Answer 10

In a Maclaurin Series, r stands for the rth recursion… where the first recursion is the 0th.

The five Maclaurin Series functions are:

(1+x)ⁿ = 1 + nx + (n(n-1)/2!)x²…(n-1+r))/r! xʳ … for -1 < x < 1, n = REAL

eˣ = 1 + x + x²/2! + … xʳ / r! … for all x

ln(1+x) = ((-1)ʳ⁺¹ x (xʳ)/r) … for -1 < x ≤ 1

sinx  = ((-1)ʳ x ((x²ʳ+1)/(2r+1)!) ... for all x
cosx = (-1)ʳ x (x²ʳ / (2r)!) ... for all x