Algebraic Methods Flashcards
What is the factor theorem?
If f(a) = 0, then (x - a) is a factor of f(x).
What is the remainder theorem?
The remainder when f(x) is divided by (x - a) is f(a).
How do you divide polynomials?
Use long division or algebraic division.
What type of divisor is used in A Level polynomial division?
Only linear divisors like (x ± a).
What is algebraic long division used for?
To simplify rational expressions or find remainders.
What is the structure of polynomial division?
Dividend = (Divisor × Quotient) + Remainder.
How do you express a rational function as a quotient and remainder?
Write it as f(x)/g(x) = Q(x) + R(x)/g(x).
What are partial fractions?
A method of expressing a rational function as a sum of simpler fractions.
When can you split a rational expression into partial fractions?
When the degree of numerator < degree of denominator.
What do you do before applying partial fractions if degree of numerator ≥ denominator?
Use algebraic division first.
What are the typical forms used in partial fractions?
A/(x+a), A/(x+a)², (Ax + B)/(quadratic).
What is the process of finding constants in partial fractions?
Multiply through and compare coefficients or substitute suitable x-values.
What is an improper fraction in algebra?
When the degree of the numerator is ≥ the degree of the denominator.
How do you simplify an improper algebraic fraction?
Divide the numerator by the denominator.
How is partial fraction decomposition useful in integration?
It simplifies complex rational functions for easier integration.
Can you use partial fractions with repeated linear factors?
Yes, include terms like A/(x + a) + B/(x + a)².
What should you do if a quadratic factor cannot be factorised?
Use a form like (Ax + B)/(quadratic).
How do you check your partial fractions?
Recombine them to verify you get the original expression.
Why are algebraic methods important?
They simplify expressions and help solve complex algebra problems.
When are algebraic methods used in A Level?
In simplifying expressions, solving equations, and calculus.
