Simplify (4x)/(x^2+3x)-5/(x+3)
The problem presented is asking you to perform algebraic simplification on a compound fraction composed of two separate fractions. The first fraction has (4x) in the numerator and a quadratic expression (x^2+3x) in the denominator. The second fraction, which is subtracted from the first, has a constant (-5) in the numerator and a linear expression (x+3) in the denominator. The overall task is to simplify this expression into a simpler form, possibly by combining terms, factoring, or by finding a common denominator.
$\frac{4 x}{x^{2} + 3 x} - \frac{5}{x + 3}$
Answer
Solution:
Step 1: Break down each term for simplification.
Step 1.1: Extract the common factor from the denominator of the first term.
Step 1.1.1: Identify $x$ as a factor in $x^{2}$.
$$\frac{4x}{x \cdot x + 3x} - \frac{5}{x + 3}$$
Step 1.1.2: Recognize $x$ as a factor in $3x$.
$$\frac{4x}{x \cdot x + x \cdot 3} - \frac{5}{x + 3}$$
Step 1.1.3: Factor out $x$ from the sum $x \cdot x + x \cdot 3$.
$$\frac{4x}{x(x + 3)} - \frac{5}{x + 3}$$
Step 1.2: Eliminate the common $x$ factor from the first term.
Step 1.2.1: Simplify by canceling out the $x$.
$$\frac{4 \cancel{x}}{\cancel{x}(x + 3)} - \frac{5}{x + 3}$$
Step 1.2.2: Present the simplified expression.
$$\frac{4}{x + 3} - \frac{5}{x + 3}$$
Step 2: Merge the fractions into a single term.
Step 2.1: Combine the numerators over a shared denominator.
$$\frac{4 - 5}{x + 3}$$
Step 2.2: Condense the expression further.
Step 2.2.1: Compute the difference between the numerators.
$$\frac{-1}{x + 3}$$
Step 2.2.2: Position the negative sign in front of the fraction.
$$-\frac{1}{x + 3}$$
Knowledge Notes:
The problem involves simplifying a rational expression, which is a fraction where the numerator and/or the denominator are polynomials. The steps taken to simplify the expression include:
Factoring: This is the process of breaking down a polynomial into a product of its factors. In this case, we factored out $x$ from the denominator $x^2 + 3x$.
Canceling common factors: When a factor appears in both the numerator and the denominator of a fraction, it can be canceled out. This simplifies the fraction.
Combining like terms: When two fractions have the same denominator, they can be combined by adding or subtracting the numerators and keeping the common denominator.
Simplifying the expression: After combining like terms, the expression may further be simplified by performing the addition or subtraction in the numerator.
In this particular problem, we applied these steps to simplify the given rational expression. The final result is a simplified fraction that is easier to understand and work with.
