FMAT pure1 roots of polynomials Flashcards
What is the general form of a quadratic equation?
The general form is ax^2 + bx + c = 0, where α and β are the roots.
What are the relationships between the roots and coefficients of a quadratic equation?
Sum of roots: α + β = -b/a
Product of roots: αβ = c/a
How do you derive the relationships between roots and coefficients in a quadratic equation?
By expressing the equation as a(x - α)(x - β) = 0, expanding, and comparing coefficients.
How do you find a quadratic equation with given roots α and β?
Use the relationships:
x^2 - (α + β)x + αβ = 0
What are the relationships between the roots and coefficients of a cubic equation ax^3 + bx^2 + cx + d = 0?
Sum of roots: α + β + γ = -b/a
Sum of products of roots in pairs: αβ + βγ + γα = c/a
Product of roots: αβγ = -d/a
What is the substitution method for finding a polynomial with transformed roots?
Replace x with a function of u, e.g., x = u + k, and substitute into the original equation to derive the new equation.
What are the relationships between the roots and coefficients of a quartic equation ax^4 + bx^3 + cx^2 + dx + e = 0?
Sum of roots: α + β + γ + δ = -b/a
Sum of products of roots in pairs: αβ + αγ + αδ + βγ + βδ + γδ = c/a
Sum of products of roots in triples: αβγ + βγδ + γδα + δαβ = -d/a
Product of roots: αβγδ = e/a
How do you find a cubic equation with roots 1 + α, 1 + β, 1 + γ?
Use u = x - 1, substitute x = u + 1 into the original cubic equation, and simplify.
What is the sum of squares of roots in a cubic equation?
α^2 + β^2 + γ^2 = (α + β + γ)^2 - 2(αβ + βγ + γα)
How do you find the sum of products of roots in pairs for a quartic equation?
Compute all unique combinations of two roots and sum them:
αβ + αγ + αδ + βγ + βδ + γδ.
How do you find a quartic equation with roots 2α, 2β, 2γ, 2δ?
Substitute z = w / 2 into the original equation and simplify to derive the new quartic equation.
How does changing the coefficient a affect a polynomial?
It scales the entire polynomial without changing the roots.
What happens to the roots of a quadratic equation when they are transformed as 2α + 1 and 2β + 1?
Replace x with u = (x - 1) / 2 to find the new equation.
