# Chapter 2 - Algebra and Series Flashcards

## Roots of Polynomials, Inequalities, Summing Series, Proof by induction and Maclaurin: this is a long series...

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?

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?

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

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?

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

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?

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.

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

n

∑r = ?

1

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

n

∑r = n(n+1) /2

1

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

n

∑r² = ?

1

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

n

∑r² = n(n+1)(2n+1)/6

1

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

n

∑r³ = ?

1

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

n

∑r³ = (n²)(n+1)² /4

1

What are the steps for Proof by Induction?

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:

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

What is the definition of a Maclaurin series?

The definition of a Maclaurin series is:

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

What are the five Maclaurin Series functions?

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