Simplify (6x^2+15x)/(8x^2+20)

The problem asks to perform a simplification of a rational algebraic expression, which is essentially a fraction consisting of a polynomial in the numerator and a polynomial in the denominator. The task is to reduce this fraction to its simplest form by factoring out any common factors that appear in both the numerator and the denominator.

$\frac{6 x^{2} + 15 x}{8 x^{2} + 20}$

Answer

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Solution:

Simplification Process

Step 1: Factor out the common term from the numerator

Extract the greatest common factor (GCF) from the terms in the numerator.

Factor out $3 x$ from $6 x^{2}$ : $\frac{3 x (2 x) + 15 x}{8 x^{2} + 20}$
Factor out $3 x$ from $15 x$ : $\frac{3 x (2 x) + 3 x (5)}{8 x^{2} + 20}$
Combine the factored terms: $\frac{3 x (2 x + 5)}{8 x^{2} + 20}$

Step 2: Factor out the common term from the denominator

Extract the GCF from the terms in the denominator.

Factor out $4$ from $8 x^{2}$ : $\frac{3 x (2 x + 5)}{4 (2 x^{2}) + 20}$
Factor out $4$ from $20$ : $\frac{3 x (2 x + 5)}{4 (2 x^{2}) + 4 (5)}$
Combine the factored terms: $\frac{3 x (2 x + 5)}{4 (2 x^{2} + 5)}$

Knowledge Notes:

To simplify a rational expression, you can factor out common terms from the numerator and the denominator. Factoring involves finding the greatest common factor (GCF) that can be divided out of each term in a polynomial.

Greatest Common Factor (GCF): The largest factor that divides two numbers or terms. For example, the GCF of $6 x^{2}$ and $15 x$ is $3 x$ .
Factoring a Polynomial: Rewriting a polynomial as a product of its factors. This can simplify expressions and solve equations.
Simplifying Rational Expressions: After factoring, if the numerator and denominator have a common factor, they can be divided by that factor to simplify the expression.

In this problem, we factored out $3 x$ from the numerator and $4$ from the denominator. This process did not lead to further simplification because the resulting terms in the numerator and denominator are not common factors that can be divided out. Therefore, the simplified form of the expression is $\frac{3 x (2 x + 5)}{4 (2 x^{2} + 5)}$ .