2.1.5 Understanding and using algebraic fractions

• What is an algebraic fraction?

  Fraction with variable expressions
• What is the numerator in the algebraic fraction (3x + 2) / 5x?

  3x + 2
• In the algebraic fraction (3x + 2) / 5x, the numerator is 3x + 2 and the denominator is 5x

  True
• What is the denominator in the algebraic fraction (3x + 2) / 5x?


  5x
• What is the first step in simplifying an algebraic fraction?

  Factorize numerator and denominator
• The algebraic fraction (4x + 8) / (2x + 4) simplifies to 2.


  True
• The smallest common denominator is also known as the lowest common multiple (LCM).

  True
• What must algebraic fractions have in order to be added or subtracted?

  Common denominator
• What is the result of multiplying the denominators (2x + 1) and (x)?

  $2x^{2} +$$x$
• When multiplying $\frac{3x}{2x + 1}$ by $\frac{x}{5x - 2}$, the result is $\frac{3x^{2}}{(2x + 1)(5x - 2)}$.


  True
• Steps to multiply $\frac{3x}{2x + 1}$ and $\frac{5x - 2}{x}$


  1️⃣ Multiply numerators: (3x)(5x - 2) = 15x^2 - 6x

  2️⃣ Multiply denominators: (2x + 1)(x) = 2x^2 + x

  3️⃣ Form the new fraction: (15x^2 - 6x) / (2x^2 + x)

  4️⃣ Simplify: (15x - 6) / (2x + 1)
• What is the numerator in an algebraic fraction?

  The expression on top
• What is the first step in dividing algebraic fractions?

  Invert the divisor
• Every algebraic fraction has two parts: a numerator and a denominator.


  True
• What should you do after multiplying the first fraction by the reciprocal of the divisor?

  Simplify the result
• What is the simplified form of \frac{3x^{2}}{10x^{2} - 4x + 5x - 2}</latex>?

  \frac{3x^{2}}{10x^{2} + x - 2}
• To divide fractions, you multiply by the reciprocal of the divisor.


  True
• What is the equation being solved in the example?

  $\frac{x}{3} +$$\frac{2x}{5} =$$2$
• Both the numerator and denominator in the algebraic fraction (3x + 2) / 5x are variable expressions


  True
• The result of $\frac{3x}{2x + 1} \div \frac{5x - 2}{x}$ is \frac{3x^{2}}{10x^{2} + x - 2}.


  True
• The final step in solving equations with algebraic fractions is to check that the solution does not make any denominators zero.


  True
• Match the algebraic fraction with its numerator and denominator:

  (2x + 3) / (4x - 5) ↔️ Numerator: 2x + 3 ||| Denominator: 4x - 5

  (7y) / 2y ↔️ Numerator: 7y ||| Denominator: 2y
• In the example, the solution $x =$$\frac{30}{11}$ is valid because it does not make any denominators zero.


  True
• What is the first step in multiplying algebraic fractions?

  Multiply the numerators
• Multiplying by the common denominator removes the fractions by canceling out the denominators.

  True
• What is the simplified equation after multiplying by the common denominator in the example?

  $11x =$$30$
• Steps to find a common denominator for algebraic fractions

  1️⃣ Multiply the denominators

  2️⃣ Simplify the result
• What is the result of multiplying the numerators (3x) and (5x - 2)?

  $15x^{2} - 6x$
• The final step in multiplying algebraic fractions is to simplify the new fraction if possible.


  True
• What is the first step when adding (2x + 3) / (3x) and (5x - 1) / (3x)?

  (2x + 3) + (5x - 1)
• The result of (2x + 3) / (3x) + (5x - 1) / (3x) is (7x + 2) / (3x).


  True
• What is the result after canceling out the common factor in (4(x + 2)) / (2(x + 2))?

  4 / 2
• Steps to simplify algebraic fractions by canceling common factors

  1️⃣ Factorize both the numerator and denominator

  2️⃣ Identify common factors between the numerator and denominator

  3️⃣ Cancel out the common factors

  4️⃣ Simplify the fraction
• What should you do when algebraic fractions have common denominators and you need to add or subtract them?

  Combine numerators
• After combining the numerators, you should always simplify the new fraction if possible.


  True