Simplify (x^2+2x-3)/(x^2-9x+8)
The problem is asking to perform algebraic simplification on the given rational expression. In the numerator, there is a quadratic polynomial (x^2 + 2x - 3), and in the denominator, there is another quadratic polynomial (x^2 - 9x + 8). The task is to simplify this complex fraction by factoring both the numerator and the denominator, if possible, and then reducing common factors to achieve the simplest form of the expression.
$\frac{x^{2} + 2 x - 3}{x^{2} - 9 x + 8}$
Answer
Solution:
Step 1: Factor the numerator $x^2 + 2x - 3$ using the AC method.
Step 1.1: Identify two integers whose product equals $c$ and whose sum equals $b$ in the quadratic $x^2 + bx + c$. For the given quadratic, find integers with a product of $-3$ and a sum of $2$. The integers are $-1$ and $3$.
Step 1.2: Express the numerator in its factored form using the identified integers: $\frac{(x - 1)(x + 3)}{x^2 - 9x + 8}$.
Step 2: Factor the denominator $x^2 - 9x + 8$ using the AC method.
Step 2.1: Find two integers whose product is $c$ and whose sum is $b$ for the quadratic $x^2 + bx + c$. For this quadratic, we need integers with a product of $8$ and a sum of $-9$. The integers are $-8$ and $-1$.
Step 2.2: Write the denominator in its factored form using these integers: $\frac{(x - 1)(x + 3)}{(x - 8)(x - 1)}$.
Step 3: Simplify the expression by canceling out common factors.
Step 3.1: Identify and cancel the common factor in the numerator and denominator: $\frac{\cancel{(x - 1)}(x + 3)}{(x - 8)\cancel{(x - 1)}}$.
Step 3.2: The simplified form of the expression is: $\frac{x + 3}{x - 8}$.
Knowledge Notes:
To simplify a rational expression, the following knowledge points are relevant:
Factoring Quadratics: This involves rewriting a quadratic expression in the form $ax^2 + bx + c$ as a product of two binomials. The AC method is a factoring technique where you look for two numbers that multiply to $a \times c$ and add up to $b$.
The AC Method: This method is used to factor quadratics where $a \neq 1$. It involves finding two numbers that multiply to give the product of the coefficient of $x^2$ (a) and the constant term (c), and add up to the coefficient of $x$ (b).
Canceling Common Factors: When a factor appears in both the numerator and the denominator of a fraction, it can be canceled out, simplifying the fraction.
Simplified Form of a Rational Expression: After canceling common factors, the expression is rewritten in its simplest form, where no further reduction is possible.
In the provided solution, the AC method is used to factor both the numerator and the denominator, and then common factors are canceled to simplify the rational expression.
