Intuition The one core ideaA big ugly fraction of polynomials was built by adding together a few simple fractions over a common denominator — and adding is reversible, so we can un-add it back into those simple pieces. Every symbol below exists to answer one of two questions: "what were the simple pieces?" or "how do I integrate each simple piece?"
Before you can trust the partial fractions machine , you need every symbol it silently assumes. We build them one at a time, from nothing, each anchored to a picture.
A polynomial is a sum of terms, each a number times a whole-number power of x x 2 − x − 2 3 x + 1 degree is the biggest power that appears. So x 2 − x − 2 2 3 x + 1 1 7 0
Think of the degree as "how curvy the graph is allowed to be": degree 1 is a straight line, degree 2 is one bend (a parabola), higher degrees wiggle more.
Definition Rational functionA rational function is one polynomial divided by another: Q ( x ) P ( x ) P numerator ) and Q denominator ) are both polynomials. The letters P Q P particular top, Q quotient's bottom.
deg deg P P deg ( 3 x + 1 ) = 1 deg P < deg Q
Intuition Why "proper" mattersLook at the figure. When the top has smaller degree than the bottom (deg P < deg Q 0 x proper . When the top is as curvy or curvier (deg P ≥ deg Q improper , and no sum of shrinking proper pieces could ever equal it. That is why the recipe demands proper input first.
To factor a polynomial is to write it as a product of smaller polynomials, like turning x 2 − x − 2 ( x − 2 ) ( x + 1 ) factor . (See Factoring polynomials over the reals .)
A root of a polynomial is a value of x 0 ( x − 2 ) x = 2 x = 2 2 − 2 = 0 ( x − a ) exactly when x = a
Definition Linear vs. quadratic factor
A linear factor has degree 1 ( x − a )
A quadratic factor has degree 2 x 2 + b x + c
Definition The discriminantFor x 2 + b x + c discriminant is the number b 2 − 4 c does this quadratic hit zero on the real number line?
b 2 − 4 c > 0 factors into two linear pieces.b 2 − 4 c = 0 b 2 − 4 c < 0 no real roots → it is irreducible (cannot be broken into real linear factors).
Worked example Discriminant in actionx 2 + 1 b = 0 , c = 1 = 0 2 − 4 ( 1 ) = − 4 < 0 x 2 − 5 x + 6 25 − 24 = 1 > 0 ( x − 2 ) ( x − 3 ) not allowed to keep its quadratic template.
Definition Repeated factor and the exponentk ( x − a ) k ( x − a ) k k exponent : ( x − 1 ) 2 = ( x − 1 ) ( x − 1 ) touches the axis and turns back instead of crossing straight through.
The parent note gives repeated factors a whole stack of fractions (x − a A 1 + ( x − a ) 2 A 2 + …
Definition Unknown numeratorsThe letters A , B , C , … placeholders for numbers we don't yet know . Partial fractions writes the answer as a template full of these unknowns, then solves for them. Subscripts like A 1 , A 2 B x + C B x C
∑ j = 1 ∑ m j 1 , 2 , … , m ∑ j = 1 3 ( x − a ) j A j = x − a A 1 + ( x − a ) 2 A 2 + ( x − a ) 3 A 3
Once a fraction is split, each piece integrates using ONE of three standard results. Here is why each tool and not another .
Definition The integral sign andd x ∫ f ( x ) d x "what function has f ( x ) The d x x + C
Definition Substitution — the glue toolIntegration by substitution lets us rename a chunk of the integrand as a single letter u u ∫ x 2 + 1 x d x = 2 1 ln ( x 2 + 1 ) u = x 2 + 1 x d x 2 1 d u
Definition Completing the squareCompleting the square rewrites x 2 + b x + c ( x + 2 b ) 2 + d 2 Why? The arctan tool needs the exact shape ( stuff ) 2 + ( constant ) 2
Intuition Numerator degree is one less than denominatorOver a common denominator, a single fraction's top can only reach a degree one below its bottom. A linear bottom ( x − a ) A B x + C why a repeated factor needs all lower powers to recover the missing freedom.
Discriminant tells irreducible
Test yourself — each line reveals the answer.
What does deg P < deg Q The numerator is less curvy than the denominator (proper); only then can shrinking simple pieces sum to it.
When is x 2 + b x + c When the discriminant b 2 − 4 c < 0
What is the discriminant of x 2 − 5 x + 6 25 − 24 = 1 > 0 ( x − 2 ) ( x − 3 )
Why does a quadratic factor get a numerator B x + C A A top can reach degree one below its bottom; a degree-2 bottom allows a degree-1 top.
Which integral gives a log, and why? ∫ x − a 1 d x = ln ∣ x − a ∣ ln x 1
Why does ∫ ( x − a ) − 2 d x − ( x − a ) − 1 The log only comes from exponent − 1
What shape must a quadratic be forced into before arctan works, and how? ( x + 2 b ) 2 + d 2 square + const 1
What does arctan "Which angle has this tangent?" — it converts a square + const 1
What does the subscript in A 1 , A 2 Which power of a repeated factor each unknown belongs to.
