We can also reverse this through a process called “partial fracti ons”.
A+3B2A−5B2A+6B2A−5Bsubtract the previous two functions0−11BB3(−111)+A=1A=3x2+x−10x+3=(3x−5)(x+2)x+3=3x−5A+x+2B=(3x−5)(x+2)A(x+2)+B(3x−5)=(3x−5)(x+2)(A+3B)x+2A−5B=1=3=2=3=−1=−111=1114=3x−51114+x+2−111We know from the first fractionmultiply by 2 so things can cancel
This is useful for finding integrals of complicated fractions. ∫3x2+x−10x+3dx=∫3x−51114−∫x+2111=1114∫3x−51−111∫x+21=3314ln∣3x−5∣−111ln∣x+2∣+C
Another example ∫x2+5xx−2dxA+B5AAB=x(x+5)x−2=xA+x+5B=x(x+5)A(x+5)+x(x+5)B(x)=x(x+5)(A+B)x+5A=1=−2=−52=57=x−52+x+557=∫x−52+∫x+557=−52∫x1+57∫x+51=−52ln∣x∣57ln∣x∣C
Review: How do I factor polynomials?
DONE
Completed: 2022-05-16
For x2+7x+12, what two numbers would add up to make 7, but multiply to make 12? 3+4.
You can also do GCF which says “is there a number which divides evenly into all terms? Is there an x which divides evenly into all terms? If so, that’s a factor so pull it out”.