Simplify Simplify the expression (9x^3+6x^2)/(3x+2)
The question asks for the simplification of a given algebraic expression, which is a rational expression in the form of a polynomial divided by a binomial. The expression to be simplified is (9x^3 + 6x^2) / (3x + 2). The task involves applying algebraic techniques such as polynomial division, factoring, or cancellation, if possible, to reduce the expression to its simplest form.
Simplify the expression$\frac{9 x^{3} + 6 x^{2}}{3 x + 2}$
Answer
Solution:
Simplification Process
Step 1: Extract the common factor
Identify and factor out the common term $3x^2$ from the numerator $9x^3 + 6x^2$.
Step 1.1
Extract $3x^2$ from $9x^3$. We get $\frac{3x^2(3x) + 6x^2}{3x + 2}$.
Step 1.2
Extract $3x^2$ from $6x^2$. We get $\frac{3x^2(3x) + 3x^2(2)}{3x + 2}$.
Step 1.3
Combine the factored terms. We obtain $\frac{3x^2(3x + 2)}{3x + 2}$.
Step 2: Simplify by canceling common factors
Eliminate the common factor $(3x + 2)$ from the numerator and denominator.
Step 2.1
Cancel out the common factor. We have $\frac{3x^2(\cancel{3x + 2})}{\cancel{3x + 2}}$.
Step 2.2
Simplify to get the final result: $3x^2$.
Knowledge Notes:
To simplify a rational expression, we can follow these steps:
Factorization: Break down both the numerator and denominator into their simplest factors. This involves identifying common terms or patterns that can be factored out.
Common Factors: Look for common factors in the numerator and denominator. These are terms that appear in both and can be canceled out to simplify the expression.
Cancellation: After factoring, cancel out the common factors. This is based on the principle that a fraction can be reduced by dividing both the numerator and the denominator by the same non-zero number or expression.
Simplify: Once the common factors are canceled, rewrite the expression in its simplest form.
In the given problem, we used the distributive property to factor out $3x^2$ from the numerator. The distributive property states that for any numbers or expressions $a$, $b$, and $c$, the equation $a(b + c) = ab + ac$ holds true. After factoring, we noticed that the term $(3x + 2)$ was present in both the numerator and the denominator, allowing us to cancel it out and simplify the expression to $3x^2$.
