Simplify (x^3+10x^2)/(x^2+5x-50)
The problem provided is asking for the simplification of a rational expression, which consists of a polynomial in the numerator (x^3 + 10x^2) and another polynomial in the denominator (x^2 + 5x - 50). Simplifying this expression typically involves factoring both the numerator and the denominator if possible, then reducing the expression by cancelling out any common factors shared between the numerator and the denominator. The goal is to write the expression in its simplest form.
$\frac{x^{3} + 10 x^{2}}{x^{2} + 5 x - 50}$
Answer
Solution:
Step:1
Extract $x^{2}$ from the numerator $x^{3} + 10x^{2}$.
Step:1.1
Take $x^{2}$ out of $x^{3}$ to get $\frac{x^{2} \cdot x + 10 \cdot x^{2}}{x^{2} + 5x - 50}$.
Step:1.2
Take $x^{2}$ out of $10x^{2}$ to obtain $\frac{x^{2} \cdot x + x^{2} \cdot 10}{x^{2} + 5x - 50}$.
Step:1.3
Combine the factored $x^{2}$ to simplify the numerator as $\frac{x^{2}(x + 10)}{x^{2} + 5x - 50}$.
Step:2
Factor the denominator $x^{2} + 5x - 50$ by the AC method.
Step:2.1
Identify two numbers whose product equals $c$ and sum equals $b$ for the quadratic $x^{2} + bx + c$. Here, find two numbers whose product is $-50$ and sum is $5$. The numbers are $-5$ and $10$.
Step:2.2
Express the denominator in its factored form using these numbers as $\frac{x^{2}(x + 10)}{(x - 5)(x + 10)}$.
Step:3
Eliminate the common factor $(x + 10)$.
Step:3.1
Cross out the common $(x + 10)$ factor to get $\frac{x^{2} \cancel{(x + 10)}}{(x - 5) \cancel{(x + 10)}}$.
Step:3.2
Write down the simplified expression as $\frac{x^{2}}{x - 5}$.
Knowledge Notes:
Factoring out common terms: When a common term appears in each term of a polynomial, it can be factored out to simplify the expression. In this case, $x^{2}$ is factored out from the numerator.
AC Method: This is a technique used to factor quadratics of the form $ax^{2} + bx + c$. The method involves finding two numbers that multiply to $ac$ and add up to $b$. These two numbers are then used to split the middle term and factor by grouping.
Canceling common factors: When the same factor appears in both the numerator and the denominator of a fraction, it can be canceled out to simplify the expression. This is based on the property that $\frac{a}{a} = 1$ for any non-zero $a$.
Simplifying Rational Expressions: The process involves factoring both the numerator and the denominator and then canceling out common factors. The result is a simplified form of the original rational expression.
Quadratic Factoring: For a quadratic expression $ax^{2} + bx + c$, if it can be factored, it will typically be in the form of $(dx + e)(fx + g)$, where $d \cdot f = a$, $e \cdot g = c$, and $d \cdot g + e \cdot f = b$.
