Simplify (y^2+yx+3y+3x)/(y+3)
The question provided is asking for a simplification of the given algebraic expression. It involves a polynomial in the form of a fraction, with the numerator being a second-degree polynomial (y^2 + yx + 3y + 3x) and the denominator being a first-degree polynomial (y + 3). The task is to perform operations to reduce this expression to its simplest form, where no further factorization or division can be carried out. This often involves factoring the numerator, canceling out common factors between the numerator and the denominator, and rearranging terms if necessary, in order to present the expression in a more compact and simplified manner.
$\frac{y^{2} + y x + 3 y + 3 x}{y + 3}$
Answer
Solution:
Step:1
Break down the numerator.
Step:1.1
Isolate common factors from each term pair.
Step:1.1.1
Separate the terms into two groups: $\frac{(y^2 + yx) + (3y + 3x)}{y + 3}$
Step:1.1.2
Extract the common factor from each pair: $\frac{y(y + x) + 3(y + x)}{y + 3}$
Step:1.2
Extract the common factor $y + x$ from the numerator: $\frac{(y + x)(y + 3)}{y + 3}$
Step:2
Eliminate the common term $y + 3$.
Step:2.1
Remove the shared term: $\frac{(y + x)(\cancel{y + 3})}{\cancel{y + 3}}$
Step:2.2
Simplify to the result: $y + x$
Knowledge Notes:
The problem involves simplifying a rational expression. The steps taken to simplify the expression are based on factoring and canceling common factors in the numerator and denominator.
Factoring: This is the process of breaking down an expression into products of other expressions or factors. In this case, the numerator is factored by grouping terms that have a common factor and then extracting that factor.
Common Factor: A common factor is a term that is present in all parts of an expression. In this problem, $y + x$ is a common factor in the numerator.
Canceling: In rational expressions, if the same term appears in both the numerator and the denominator, they can be canceled out, as they would divide to one.
Simplification: After canceling, the expression is simplified to its lowest terms, which is the final answer.
Latex Formatting: The use of Latex in the solution helps to clearly display mathematical expressions and operations. For example, $\frac{y(y + x) + 3(y + x)}{y + 3}$ shows the factored form of the numerator over the denominator.
Rational Expressions: A rational expression is a fraction where both the numerator and the denominator are polynomials. Simplifying rational expressions often involves factoring and canceling common polynomial factors.
By understanding these concepts, one can effectively simplify rational expressions and solve similar problems.
