(c+6b)/(ac+2bc-6ab-3a^(2))+(2b)/(a^(2)+2ab)-(b)/(ac-3a^(2))

Jasonbradley

3 min read

·

Jun 16, 2024

59

Share

Simplifying the Expression: (c+6b)/(ac+2bc-6ab-3a^(2))+(2b)/(a^(2)+2ab)-(b)/(ac-3a^(2))

This article will guide you through the steps to simplify the given algebraic expression:

(c+6b)/(ac+2bc-6ab-3a^(2))+(2b)/(a^(2)+2ab)-(b)/(ac-3a^(2))

Step 1: Factor the Denominators

• Denominator 1: ac+2bc-6ab-3a^(2)

  • Factor by grouping: (ac+2bc) - (6ab+3a^(2)) = c(a+2b) - 3a(2b+a) = (c-3a)(a+2b)

• Denominator 2: a^(2)+2ab

  • Factor out a common factor: a(a+2b)

• Denominator 3: ac-3a^(2)

  • Factor out a common factor: a(c-3a)

Now our expression becomes:

(c+6b)/((c-3a)(a+2b)) + (2b)/(a(a+2b)) - (b)/(a(c-3a))

Step 2: Find the Least Common Multiple (LCM) of the Denominators

The LCM of the denominators is a(c-3a)(a+2b).

Step 3: Rewrite each fraction with the LCM as the denominator

• Fraction 1: (c+6b)/((c-3a)(a+2b)) * (a/a) = (ac+6ab)/(a(c-3a)(a+2b))

• Fraction 2: (2b)/(a(a+2b)) * ((c-3a)/(c-3a)) = (2bc-6ab)/(a(c-3a)(a+2b))

• Fraction 3: (b)/(a(c-3a)) * ((a+2b)/(a+2b)) = (ab+2b^(2))/(a(c-3a)(a+2b))

Step 4: Combine the Fractions

Now we have:

(ac+6ab)/(a(c-3a)(a+2b)) + (2bc-6ab)/(a(c-3a)(a+2b)) - (ab+2b^(2))/(a(c-3a)(a+2b))

Since the denominators are now the same, we can combine the numerators:

(ac+6ab + 2bc - 6ab - ab - 2b^(2))/(a(c-3a)(a+2b))

Step 5: Simplify the Numerator

** (ac + 2bc - ab - 2b^(2))/(a(c-3a)(a+2b)) **

Final Simplified Expression

The simplified form of the given expression is:

(ac + 2bc - ab - 2b^(2))/(a(c-3a)(a+2b))