Pure

• What is the main concept behind manipulating algebraic fractions?

  Algebraic fractions can be manipulated in the same way as numeric fractions.
• How do you add the algebraic fractions $\frac{x}{x + 4} +$$\frac{4}{x - 1}$?


  \frac{x(x - 1) + 4(x + 4)}{(x + 4)(x - 1)} =$\frac{x^2 + 3x + 16}{(x + 4)(x - 1)}$
• What is the result of multiplying the algebraic fractions $\frac{x}{x + 4} \times \frac{4}{x - 1}$?


  $\frac{4x}{(x - 1)(x + 4)}$
• How do you divide the algebraic fractions $\frac{x}{x + 4} \div \frac{4}{x - 1}$?


  $\frac{x(x - 1)}{4(x + 4)}$
• What is the relationship between a fraction of the form $\frac{F(x)}{G(x)}$ and its quotient and remainder?


  $\frac{F(x)}{G(x)} =$$Q(x) +$$\frac{r}{G(x)}$, where $Q(x)$ is the quotient and $r$ is the remainder.
• How can an improper fraction be expressed using algebraic division?

  An improper fraction can be rewritten as a proper fraction plus a quotient.
• What must you do before using partial fractions on an improper fraction?


  You must first perform long division to express it as a proper fraction.
• What is a linear factor in the context of partial fractions?

  A linear factor is of the form $ax +$$b$.
• What is an improper fraction?

  An improper fraction is one where the degree of the numerator is greater than or equal to the degree of the denominator.
• What is proof by contradiction?

  Proof by contradiction assumes a statement is false and shows that this leads to a contradiction.
• How can any even number $n$ be expressed?


  Any even number $n$ can be written as $n =$$2k$ for some integer $k$.
• How can any odd number $n$ be expressed?


  Any odd number $n$ can be written as $n =$$2k +$$1$ for some integer $k$.
• How can rational numbers be expressed?

  Rational numbers can be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers.
• How can irrational numbers be expressed?

  Irrational numbers cannot be written in the form $\frac{a}{b}$.
• What is the process of proving statements by contradiction?

  1. Assume the statement is false.
  2. Show that this assumption leads to a contradiction.
  3. Conclude that the original statement must be true.

  
• What are the steps to prove that there are infinitely many prime numbers by contradiction?


  1. Assume there are a finite number of primes.
  2. Define $K =$$p_1 p_2 p_3 \ldots p_n +$$1$.
  3. Show $K$ is not divisible by any of the primes.
  4. Conclude that there must be infinitely many primes.
• What are the steps to prove that there exist no integers a and b such that $21a +$$14b =$$1$?


  1. Assume integers a and b exist such that $21a +$$14b =$$1$.
  2. Divide through by 7 to get $3a +$$2b =$$1$.
  3. Show that $3a +$$2b$ must also be an integer.
  4. Conclude that the assumption is false.
• What are the important definitions related to fractions and statements?

  • Negation: A statement implying the original statement is incorrect.
  • Contradiction: Incompatibility between two statements.
  • Improper fraction: Degree of numerator ≥ degree of denominator.

  
• What is the first example used to prove by contradiction in the study material?

  Prove that there are infinitely many prime numbers.
• What assumption is made in the first example regarding prime numbers?

  That there are a finite number of prime numbers.
• What is the significance of the number $K =$$p_1 p_2 p_3 \ldots p_n +$$1$ in the proof of infinite primes?


  It is not divisible by any of the prime numbers, leading to a contradiction.
• What conclusion can be drawn from the contradiction in the first example about prime numbers?

  There must be an infinite number of prime numbers.
• What is the second example used to prove by contradiction in the study material?

  Prove that there exist no integers a and b such that $21a +$$14b =$$1$.
• What assumption is made in the second example regarding integers a and b?


  That there are integers a and b such that $21a +$$14b =$$1$.
• What happens when we divide the equation $21a +$$14b =$$1$ by 7?


  It simplifies to $3a +$$2b =$$\frac{1}{7}$.
• Why is the equation $3a +$$2b =$$\frac{1}{7}$ a contradiction?


  Because $3a +$$2b$ must be an integer, but $\frac{1}{7}$ is not.
• What is the conclusion drawn from the second example regarding integers a and b?

  There are no integers a and b such that $21a +$$14b =$$1$.
• What are the steps involved in the partial fraction method as illustrated in Example 3?


  1. Set up the equation: $\frac{6x^2 + 5x - 2}{x(x-1)(2x+1)} \equiv \frac{A}{x} +$$\frac{B}{x-1} +$$\frac{C}{2x+1}$
  2. Make all denominators the same.
  3. Equate the numerators.
  4. Use substitution to find constants A, B, and C.
• How do you find the constants A, B, and C in the partial fraction decomposition?

  By substituting specific values of x to simplify the equation.
• What values of x are chosen for substitution to find constants in Example 3?

  x = 1 and x = 0.
• What is the result of substituting x = 1 in the equation to find B?

  It leads to $3B =$$9 \Rightarrow B =$$3$.
• What is the result of substituting x = 0 in the equation to find A?

  It leads to $-A =$$-2 \Rightarrow A =$$2$.
• How do you find the constant C after determining A and B?

  By substituting another value of x into the equation.
• What is the final result of the partial fraction decomposition in Example 3?


  $\frac{6x^2 + 5x - 2}{x(x-1)(2x+1)} \equiv \frac{2}{x} +$$\frac{3}{x-1} - \frac{4}{2x+1}$.
• What modification is needed when dealing with repeated linear factors in partial fractions?

  • Include an extra fraction for each repeated linear factor.
  • Example: For $\frac{2x^2 + 2x - 18}{x(x-3)^2}$, use:

  $\frac{A}{x} +$$\frac{B}{x-3} +$$\frac{C}{(x-3)^2}$.
• How would you set up the partial fraction decomposition for $\frac{10x^2 - 10x + 17}{(2x + 1)(x - 3)^2}$?


  $\frac{A}{2x + 1} +$$\frac{B}{x - 3} +$$\frac{C}{(x - 3)^2}$.
• How would you set up the partial fraction decomposition for $\frac{2x}{(x + 2)^2}$?


  $\frac{A}{x + 2} +$$\frac{B}{(x + 2)^2}$.
• What are the basic algebraic operations mentioned in the cheat sheet?

  • Addition
  • Multiplication
  • Division
• What is the polynomial given in the division example?

  $x^3 +$$x^2 +$$0x - 7$.
• What is the divisor in the division example?

  $x - 3$.