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أدلة الدراسة > College Algebra

Key Concepts & Glossary

Key Concepts

  • Decompose P(x)Q(x)\frac{P\left(x\right)}{Q\left(x\right)} by writing the partial fractions as Aa1x+b1+Ba2x+b2\frac{A}{{a}_{1}x+{b}_{1}}+\frac{B}{{a}_{2}x+{b}_{2}}. Solve by clearing the fractions, expanding the right side, collecting like terms, and setting corresponding coefficients equal to each other, then setting up and solving a system of equations.
  • The decomposition of P(x)Q(x)\frac{P\left(x\right)}{Q\left(x\right)} with repeated linear factors must account for the factors of the denominator in increasing powers.
  • The decomposition of P(x)Q(x)\frac{P\left(x\right)}{Q\left(x\right)} with a nonrepeated irreducible quadratic factor needs a linear numerator over the quadratic factor, as in Ax+Bx+C(ax2+bx+c)\frac{A}{x}+\frac{Bx+C}{\left(a{x}^{2}+bx+c\right)}.
  • In the decomposition of P(x)Q(x)\frac{P\left(x\right)}{Q\left(x\right)}, where Q(x)Q\left(x\right) has a repeated irreducible quadratic factor, when the irreducible quadratic factors are repeated, powers of the denominator factors must be represented in increasing powers as

    Ax+B(ax2+bx+c)+A2x+B2(ax2+bx+c)2++Anx+Bn(ax2+bx+c)n\frac{Ax+B}{\left(a{x}^{2}+bx+c\right)}+\frac{{A}_{2}x+{B}_{2}}{{\left(a{x}^{2}+bx+c\right)}^{2}}+\cdots \text{+}\frac{{A}_{n}x+{B}_{n}}{{\left(a{x}^{2}+bx+c\right)}^{n}}.

Glossary

partial fractions
the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression
partial fraction decomposition
the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions

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