12. Algebra 2

How to split a fraction into partial fractions when the denominator has a repeated root use the example 1/(x+3)(x+1)²

A/(x+3) + B/(x+1) + C/(x+1)²

How do you split 3x/(x-1)(x² + 9x - 3) into partial fractions Where the quadratic factor cannot be factorised

A/x-1 + (Bx + C)/(x² + 9x -3)

How to check partial fractions on the calculator on the big calculator

Set the partial fraction equal to the original fraction

Execute the calculation

If the output is 1 then the partial fraction is correct

If the output is 0 the partial fraction is wrong

How to check partial fractions on any calculator

The original fraction subtract the partial fractions

Answer = 0 then correct, else incorrect

What is the order of the non fractional part of the partial fraction in the case that the original fraction is top heavy

The order is the difference in the order of the numerator and denominator

Methods you can use in Alevel maths to integrate fractions

1) natural log when the top of the fraction is a multiple of the derivative of the bottom of the fraction

2) reverse chain rule → when the bottom of the fraction is to the power of 2 or more

What is the turning point of a modulus function called

Vertex

How to sketch y = |f(x)|

Draw normally

Reflect the parts of the curve that go bellow the x axis in the x axis

Such that all y values are positive

How to sketch y = f(|x|)

Sketch y = f(x) where x > 0

Reflect the the graph in the y axis

Steps to a proof by contradiction