13 Partial fractions · g10-13-01 · 5–10 min
Two denominators, one identity
How can substituting a forbidden value still find a coefficient?
Predict: Decompose (mx+n)/[(x−a)(x−b)]; predict what remains in the numerator identity at x=a.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
First root a = 0; Second root b = -1; Numerator coefficient m = 0; Numerator constant n = 1 | |
Case 2
First root a = 1; Second root b = -1; Numerator coefficient m = 1; Numerator constant n = -1 | |
Case 3
First root a = 1; Second root b = 1; Numerator coefficient m = 1; Numerator constant n = 2 | |
Revise: My claim needs the condition… because…
Exit: Decompose 1/[x(x+1)].
13 Partial fractions · g10-13-02 · 5–10 min
Why include every repeated power?
Can one term represent every numerator over (x−a)²?
Predict: Compare A/(x−a)+B/(x−a)² with the incomplete proposal B/(x−a)².
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Repeated root a = 1; Numerator coefficient m = 1; Numerator constant n = 0 | |
Case 2
Repeated root a = 1; Numerator coefficient m = 0; Numerator constant n = 2 | |
Case 3
Repeated root a = 1; Numerator coefficient m = 1; Numerator constant n = -1 | |
Revise: My claim needs the condition… because…
Exit: Decompose x/(x−1)².
13 Partial fractions · g10-13-03 · 5–10 min
Divide before decomposing
Where does the polynomial part come from?
Predict: Compare the degrees of x²+px+q and (x−a)(x−b); predict the missing polynomial term.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Numerator coefficient p = 2; Numerator constant q = 3; First root a = 1; Second root b = -1 | |
Case 2
Numerator coefficient p = 0; Numerator constant q = -1; First root a = 1; Second root b = -1 | |
Case 3
Numerator coefficient p = 0; Numerator constant q = 0; First root a = 0; Second root b = 1 | |
Revise: My claim needs the condition… because…
Exit: Rewrite x²/[x(x−1)] as a polynomial plus simple fractions.
13 Partial fractions · g10-13-04 · 5–10 min
Does cancellation repair a hole?
Are equal values on a shared domain the same as equal domains?
Predict: Compare (x−a)(x+c)/[(x−a)(x−b)] with (x+c)/(x−b), especially at x=a.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Cancelled root a = 1; Remaining denominator root b = -1; Numerator shift c = 2; Probe input x = 0 | |
Case 2
Cancelled root a = 1; Remaining denominator root b = -1; Numerator shift c = 2; Probe input x = 1 | |
Case 3
Cancelled root a = 1; Remaining denominator root b = -1; Numerator shift c = 1; Probe input x = -1 | |
Revise: My claim needs the condition… because…
Exit: Is (x²−1)/(x−1) the same function as x+1 on R?
