# Pre Calculus 11 Week 15: Adding and Subtracting Rational Expressions With Binomial and Trinomial Denominators

This week in Pre calculus 11 we learned how to add and subtract rational expressions that contained a binomial or trinomial denominator. These are quite ugly and difficult and require focus. I will show an example of each, and hopefully help you understand how to simplify these kinds of expressions.

Steps: Binomial Denominator

1. Simplify the Denominator if possible
2. Find the Common Denominator
3. Make Equivalent Fractions
5. Simplify/Reduce
6. State the Non-Permissible Values

Example: $\frac{8}{6x+9} + \frac{3}{4x-4}$

Step 1: Simplify the Denominator

•  $\frac{8}{6x+9} + \frac{3}{4x-4}$
• $\frac{8}{3(2x+3)} + \frac{3}{4(x-1)}$

Step 2: Find the Common Denominator

• $3(2x+3)\cdot 4(x-1)$
• 12(2x+3)(x-1) = Common Denominator

Step 3: Make Equivalent Fractions

Multiply the denominator of one side to the other side (top and bottom) and then use the other denominator and multiply it to the other expression. Only multiply by what is need to get to the common denominator.

• $\frac{8}{3(2x+3}\cdot\frac{4(x-1)}{4(x-1)} + \frac{3}{4(x-1)}\cdot\frac{3(2x+3)}{3(2x+3)}$
• $\frac{32(x-1)}{12(2x+3)(x-1)} + \frac{9(2x+3)}{12(2x+3)(x-1)}$
• $\frac{32x-32}{12(2x+3)(x-1)} + \frac{18x+27}{12(2x+3)(x-1)}$

Step 4: Add or Subtract                                                                                                    $\star$ Make into one big fraction

• $\frac{32x-32 + 18x+27}{12(2x+3)(x-1)}$

Step 5: Simplify/Reduce

• $\frac{32x - 32 + 18x + 27}{12( 2x + 3 )( x - 1)}$
• $\frac{32x + 18x - 32 + 27}{12( 2x + 3 )( x - 1)}$
• $\frac{50x - 5 }{12( 2x + 3 )( x - 1)}$

$\star$ Simplify the Numerator

• $\frac{50x - 5 }{12( 2x + 3 )( x - 1)}$
• $\frac{5(10x - 1) }{12( 2x + 3 )( x - 1)}$

Step 6: Non-Permissible Values

• $\frac{5(10x - 1) }{12( 2x + 3 )( x - 1)}$
• 12(2x+3)(x-1)
• 2x + 3 = 0
• 2x = -3
• $x = -\frac {3}{2}$
• $x\neq -\frac {3}{2} , 1$

FINAL ANSWER:  $\frac{5(10x - 1) }{12( 2x + 3 )( x - 1)}$

This is how you would simplify a rational expression with a binomial denominator

Steps: Trinomial Denominator

1. Simplify the Denominator if possible
2. Find the Common Denominator
3. Make Equivalent Fractions
5. Simplify/Reduce
6. State the Non-Permissible Values

Example: $\frac{b}{b^2 + 10b + 24} + \frac{2b}{b^2 + 12b + 32}$

Step 1: Simplify the Denominator

• $\frac{b}{b^2 + 10b + 24} + \frac{2b}{b^2 + 12b + 32}$
• $\frac{b}{(b + 6)(b + 4)} + \frac{2b}{(b + 8)(b+4)}$

Step 2: Find the Common Denominator

$\star$ We know that we need a (b+8), a (b+4) and a (b+6) we don’t use the extra (b+4) because there is already one being used, we don’t want have any duplicates.

• (b+8)(b+6)(b+4) Common Denominator

Step 3: Make Equivalent Fractions

Only multiply by what is needed to get to the common denominator

• $\frac{b}{(b + 6)(b + 4)}\cdot\frac {b+8}{b+8} + \frac{2b}{(b + 8)(b + 4)}\cdot\frac {b+6}{b+6}$
• $\frac{b(b+8)}{(b + 8)(b + 6)(b+4)} + \frac{2b(b+6)}{(b + 8)(b+6)(b+4)}$
• $\frac{b^2+8b}{(b + 8)(b + 6)(b+4)} + \frac{2b^2+12b}{(b + 8)(b+6)(b+4)}$

Step 4: Add or Subtract                                                                                                    $\star$ Make into one big fraction

• $\frac{b^2+8b}{(b + 8)(b + 6)(b+4)} + \frac{2b^2+12b}{(b + 8)(b+6)(b+4)}$
• $\frac{b^2+8b + 2b^2+12b}{(b + 8)(b+6)(b+4)}$

Step 5: Simplify/Reduce

• $\frac{b^2+8b + 2b^2+12b}{(b + 8)(b+6)(b+4)}$
• $\frac{b^2 + 2b^2+ 8b+ 12b}{(b + 8)(b+6)(b+4)}$
• $\frac{3b^2+ 20b}{(b + 8)(b+6)(b+4)}$

$\star$ Simplify the Numerator

• $\frac{3b^2+ 20b}{(b + 8)(b+6)(b+4)}$
• $\frac{b(3+ 20)}{(b + 8)(b+6)(b+4)}$

Step 6: Non-Permissible Values

• $\frac{b(3+ 20)}{(b + 8)(b+6)(b+4)}$
• $x\neq -8, -6, -4$

This is how you would simplify Binomial and Trinomial Rational expressions when adding and or subtracting. I hope this blog post really helped you understand how to simplify these expressions. Feel free to look back at some of my other blog posts. They may help you with anything you are struggling with. Thanks 🙂

Below are two videos to help further understanding

here is a video that helped me understand how to add and subtract (binomial denominator) kinds of expressions.

And here is another video that helped me understand how to add and subtract rational expressions with a trinomial denominator (in the video the trinomial has already been factored).