Simplify (5m^3+5m^2)/(15m^2+10m)
The problem you've presented is an algebraic expression simplification task. You are asked to simplify the given fraction, where both the numerator and the denominator are polynomial expressions involving the variable 'm'. The numerator is '5m^3 + 5m^2', which is the sum of two terms, and the denominator is '15m^2 + 10m', also the sum of two terms. The goal is to express this fraction in its simplest form by factoring out common factors, cancelling out like terms if possible, and reducing the expression to the lowest terms. This typically involves applying the distributive property and the cancellation of common terms involving the variable 'm'.
$\frac{5 m^{3} + 5 m^{2}}{15 m^{2} + 10 m}$
Answer
Solution:
Step 1: Extract common factor from numerator
Extract $5m^2$ from the numerator expression $5m^3 + 5m^2$.
Step 1.1
Extract $5m^2$ from $5m^3$. Result: $\frac{5m^2(m) + 5m^2}{15m^2 + 10m}$
Step 1.2
Extract $5m^2$ from $5m^2$. Result: $\frac{5m^2(m) + 5m^2(1)}{15m^2 + 10m}$
Step 1.3
Combine the extracted terms. Result: $\frac{5m^2(m + 1)}{15m^2 + 10m}$
Step 2: Extract common factor from denominator
Extract $5m$ from the denominator expression $15m^2 + 10m$.
Step 2.1
Extract $5m$ from $15m^2$. Result: $\frac{5m^2(m + 1)}{5m(3m) + 10m}$
Step 2.2
Extract $5m$ from $10m$. Result: $\frac{5m^2(m + 1)}{5m(3m) + 5m(2)}$
Step 2.3
Combine the extracted terms. Result: $\frac{5m^2(m + 1)}{5m(3m + 2)}$
Step 3: Cancel the common factor of $5$
Step 3.1
Eliminate the common factor of $5$. Result: $\frac{\cancel{5}m^2(m + 1)}{\cancel{5}m(3m + 2)}$
Step 3.2
Simplify the expression. Result: $\frac{m^2(m + 1)}{m(3m + 2)}$
Step 4: Cancel the common factor of $m$
Step 4.1
Factor $m$ from $m^2(m + 1)$. Result: $\frac{m(m(m + 1))}{m(3m + 2)}$
Step 4.2: Cancel the common factors
Step 4.2.1
Eliminate the common factor of $m$. Result: $\frac{\cancel{m}(m(m + 1))}{\cancel{m}(3m + 2)}$
Step 4.2.2
Simplify the expression to the final result: $\frac{m(m + 1)}{3m + 2}$
Knowledge Notes:
To simplify a rational expression, we follow these steps:
Factorization: Break down both the numerator and the denominator into their factors. This often involves factoring out the greatest common factor (GCF) or applying other factoring techniques such as factoring by grouping, using special products (difference of squares, perfect square trinomials, etc.), or factoring quadratic expressions.
Cancellation: Once the numerator and denominator are factored, cancel out any common factors that appear in both. This step simplifies the expression by reducing it to its lowest terms.
Simplification: After canceling, rewrite the expression in its simplest form. If there are no common factors left, the expression is already in its simplest form.
Algebraic Manipulation: Sometimes, additional algebraic manipulation is required to simplify the expression further. This can include expanding products or combining like terms.
In the given problem, we applied these steps by first factoring out the greatest common factor from both the numerator and the denominator, then canceling out the common factors, and finally rewriting the simplified expression. The use of Latex in the solution helps to clearly present mathematical expressions and steps.
