Cards In This Deck

• $\text{What is the first step to expand }g(x)=\frac{4-5x}{(1+x)(2-x)}\text{ using the binomial theorem?}$
  $$\begin{gathered} \text{1. Express }g(x)\text{ as partial fractions.} \\ \text{2. Rewrite the partial fractions in index form, e.g., }A(1+x{)}^{-1}+B(2-x{)}^{-1}\text{.} \\ \text{3. Expand each term separately.} \end{gathered}$$
• $$\begin{gathered} \text{True or False? } \\ \text{To simplify }\frac{x^2-9}{x^2+9x+18}\text{, you should immediately use partial fractions.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ You should factorise and simplify first.} \\ \frac{(x-3)(x+3)}{(x+3)(x+6)}=\frac{x-3}{x+6} \end{gathered}$$
• $$\begin{gathered} \text{Which method is generally more efficient for finding constants in partial fractions when the denominator contains distinct linear factors (e.g., }(x-1)(x+3)\text{)?} \\ \text{A) Equating Coefficients} \\ \text{B) Substitution} \end{gathered}$$
  $\text{B) Substitution}$
• $$\begin{gathered} \text{True or False? } \\ \text{The partial fraction form for }\frac{3}{(1-x)(1+x)}\text{ is }\frac{A}{1-x}+\frac{B}{1+x} \end{gathered}$$
  $\text{✅ True}$
• $$\begin{gathered} \text{In the identity }2x^2+3\equiv A(x-1{)}^2+B(x+1)(x-1)+C(x+1)\text{, finding }A\text{ and }C\text{ is easy by substitution (}x=-1\text{ and }x=1\text{). } \\ \text{What is the most efficient method to find constant }B\text{?} \end{gathered}$$
  $$\begin{gathered} \text{Equate the coefficients of }{x}^2\text{.} \\ 2x^2=A{x}^2+B{x}^2⟹2=A+B \end{gathered}$$
• $$\begin{gathered} \text{True or False? } \\ \text{The partial fraction decomposition of }\frac{5x}{(x-1)(x+2{)}^2}\text{ is in the form: } \\ \frac{A}{x-1}+\frac{B}{(x+2{)}^2} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Form: } \\ \frac{A}{x-1}+\frac{B}{x+2}+\frac{C}{(x+2{)}^2} \end{gathered}$$
• $$\begin{gathered} \text{What is the correct form for the partial fraction decomposition of:} \\ \frac{5x+1}{(x+1)(x-2{)}^2}\text{?} \end{gathered}$$
  $\frac{A}{x+1}+\frac{B}{x-2}+\frac{C}{(x-2{)}^2}$
• $\text{When decomposing an improper fraction where the numerator is a cubic (}{x}^3\text{) and the denominator is a quadratic (}{x}^2\text{), what form should the polynomial part take?}$
  $$\begin{gathered} Ax+B \\ \text{(A linear polynomial)} \end{gathered}$$
• $$\begin{gathered} \text{True or False? } \\ \text{An improper fraction can be defined as a fraction where the degree of the numerator is strictly greater than the degree of the denominator.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ It includes cases where the degrees are equal.} \end{gathered}$$
• $$\begin{gathered} \text{You are asked to split }\frac{3x^2+2x-1}{(x+1)(x-2)}\text{ into partial fractions. } \\ \text{What is the crucial first check you must make before setting up the form }\frac{A}{x+1}+\frac{B}{x-2}\text{?} \end{gathered}$$
  $$\begin{gathered} \text{Check the degrees of the numerator and denominator.} \\ \text{Here, the degree of the numerator (2) is equal to the degree of the denominator (2), so it is an improper fraction. You must divide first or add a constant term.} \end{gathered}$$
• $$\begin{gathered} \text{First step to integrate a rational function where the degree of the numerator is }\ge \text{ the degree of the denominator?} \\ \text{Example: }\int \frac{x^2}{x-1} \\ ,dx \end{gathered}$$
  $\text{Use Algebraic Division (Long Division) first.}$