Partial Fractions: Numerator vs Denominator | Explained in 5:30

Click For Summary
SUMMARY

In partial fraction decomposition, the degree of the numerator (N(x)) must be less than the degree of the denominator (D(x)) by at least one. If N(x) has a degree equal to or greater than D(x), polynomial division is required to simplify the expression into a proper form. This ensures that the resulting fraction can be expressed as a sum of simpler fractions, where the numerator's degree is strictly lower than that of the denominator. The leading coefficient of the numerator may be zero, but it must still adhere to this degree constraint for accurate decomposition.

PREREQUISITES
  • Understanding of polynomial functions and their degrees
  • Familiarity with polynomial long division
  • Knowledge of partial fraction decomposition techniques
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study polynomial long division methods in detail
  • Explore advanced partial fraction decomposition techniques
  • Learn about the implications of leading coefficients in polynomial expressions
  • Practice problems involving partial fractions with varying degrees
USEFUL FOR

Students in algebra or calculus courses, educators teaching polynomial functions, and anyone looking to deepen their understanding of partial fraction decomposition in mathematical analysis.

therajinator
Messages
6
Reaction score
0
Why in partial fractions does the power of the denominator have to be one more than that of the numerator, when splitting up the expression. Skip to 5:30. Thanks.
 
Last edited by a moderator:
Mathematics news on Phys.org


If the numerator (N(x)) power were greater than or equal to that of the denominator (D(x)) then you could do a polynomial division to obtain N(x)/D(x) = P(x) + Q(x)/D(x), where Q has lower degree than N.
The numerator therefore has a lower degree than the denominator.
In general, it can have any degree in that range. For the purposes of calculating it, you allow it to be up to one degree less than the denominator. The coefficient of the leading term might turn out to be zero.
 

Similar threads

High School Understanding the relationship between ratios and fractions
  • · Replies 8 ·
Replies
8
Views
2K
MHB Rationalize Numerator & Denominator
  • · Replies 2 ·
Replies
2
Views
2K
MHB Partial fractions (5x^2+1)/[(3x+2)(x^2+3)]
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Why Can't x/(x+1)^2 Be Split Like x/((x+1)(x+2))?
  • · Replies 2 ·
Replies
2
Views
2K
Partial Fractions: Why Does (x+1)2(2x+1) Need 3 Terms?
  • · Replies 7 ·
Replies
7
Views
2K
Why Do We Multiply Numerator & Denominator When Multiplying Fractions?
  • · Replies 8 ·
Replies
8
Views
17K
What's the logic behind partial fraction decomposition?
  • · Replies 2 ·
Replies
2
Views
2K
A doubt in Partial fraction decomposition
  • · Replies 8 ·
Replies
8
Views
2K
Polynomial fractions simplest form?
  • · Replies 10 ·
Replies
10
Views
4K
High School Fractional/decimal powers
  • · Replies 10 ·
Replies
10
Views
2K