Cards In This Deck

• $$\begin{gathered} \text{What is the first step to simplify the division of two algebraic fractions?} \\ \frac{a}{b}÷\frac{c}{d} \end{gathered}$$
  $$\begin{gathered} \text{Multiply the first fraction by the reciprocal of the second fraction.} \\ \frac{a}{b}\times \frac{d}{c} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{When dividing a cubic polynomial by a quadratic polynomial (e.g., }\frac{x^3}{x^2+1}\text{), the remainder will always be a constant.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{The remainder can be a linear expression (degree 1), e.g., }Ax+B\text{.} \\ \text{The degree of the remainder is always less than the degree of the divisor.} \end{gathered}$$
• $$\begin{gathered} \text{What is the first step to add these fractions?} \\ \frac{3}{x+1}-\frac{4x}{x^2-1} \end{gathered}$$
  $$\begin{gathered} \text{Factorise the denominator of the second fraction.} \\ {x}^2-1=(x+1)(x-1) \\ \text{Then find the Lowest Common Multiple (LCM), which is }(x+1)(x-1)\text{.} \end{gathered}$$
• $\text{What must you do to an improper algebraic fraction before expressing it in partial fractions?}$
  $$\begin{gathered} \text{Convert it into a mixed fraction (polynomial + proper fraction).} \\ \text{Use algebraic long division or the identity method.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \frac{2x+1}{3}-\frac{x-2}{3}=\frac{2x+1-x-2}{3}=\frac{x-1}{3} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: }\frac{x+3}{3} \\ \text{The negative sign applies to the entire numerator of the second fraction. It should be:} \\ \frac{(2x+1)-(x-2)}{3}=\frac{2x+1-x+2}{3} \end{gathered}$$
• $\text{Is the fraction }\frac{x^2+3}{x^2-1}\text{ a proper or improper algebraic fraction?}$
  $$\begin{gathered} \text{Improper.} \\ \text{An algebraic fraction is improper if the degree of the numerator is greater than or equal to the degree of the denominator. Here, both have degree 2.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \frac{x^2+5x}{x^2+2}=\frac{5x}{2} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct Answer: The expression cannot be simplified further by simple cancellation.} \\ \text{You can only cancel factors (terms multiplied together), not individual terms added or subtracted. }{x}^2\text{ is a term, not a common factor of the entire numerator and denominator.} \end{gathered}$$
• $\text{When using the Identity Method to convert an improper fraction like }\frac{x^3+x^2-7}{x-3}\text{ into mixed form, what equation do you set up?}$
  $$\begin{gathered} x^3+x^2-7\equiv (A{x}^2+Bx+C)(x-3)+D \\ \text{(Numerator) }\equiv \text{ (Quotient }\times \text{ Divisor) }+\text{ Remainder} \end{gathered}$$