Algebraic Methods

• What are fractions called when both the numerator and denominator are algebraic expressions?

  Algebraic fractions
• To simplify algebraic fractions, you must cancel common factors
• Steps to simplify the algebraic fraction \frac{8x^{3} - 4x^{2} + 6x}{2x}


  1️⃣ Divide each term in the numerator by 2x

  2️⃣ Write the simplified expression
• What is the simplified form of \frac{8x^3 - 4x^2 + 6x}{2x}</latex>?

  $4x^{2} - 2x +$$3$
• When simplifying $\frac{(x + 4)(3x - 1)}{(3x - 1)}$, cancel the common factor of (3x-1)
• What is the simplified form of $\frac{(x + 4)(3x - 1)}{(3x - 1)}$?


  x+4
• A polynomial must have positive whole number indices.

  True
• Long division can be used to divide a polynomial by $(x ± p)$, where p is a constant
• What is the result of dividing $4x^{3} +$$9x^{2} - 3x - 10$ by $(x + 2)$?


  $(x + 2)(4x^{2} +$$x - 5)$
• The factor theorem states that if f(p) = 0, then (x - p) is a factor of f(x).

  True
• To determine if a quadratic equation has real roots, find the discriminant
• A mathematical statement that has been proven is called a theorem.

  True
• What type of equation is 3x^2 + 6 = 0</latex>?

  Quadratic
• Steps to prove a mathematical statement:

  1️⃣ Start with known facts or theorems

  2️⃣ Show logical steps

  3️⃣ State the proof
• Identical statements are always equal mathematically
• What is the result of factoring $n^{2} - n$?


  $n(n - 1)$
• Match the steps of proving $(x + y)(x - y) ≡ x^{2} - y^{2}$:


  Expand $(x + y)(x - y)$ ↔️ $x^{2} - xy +$$xy - y^{2}$

  Simplify the expression ↔️ $x^{2} - y^{2}$
• What method is used to prove that the sum of two consecutive square numbers between 1^2 and 8^2 is odd?

  Proof by exhaustion
• The inequality p + q > \sqrt{4pq}</latex> holds true when p and q are both negative.

  False
• A counter-example is used to disprove a mathematical statement