Factor f(x)=(x^2-19x+88)/(x^2-121)

Solution:

Step:1

Apply the AC method to factor the numerator $x^2-19x+88$.

Step:1.1

Identify two numbers that multiply to give $c=88$ and add up to $b=-19$. The numbers are $-11$ and $-8$.

Step:1.2

Express the numerator in its factored form using the found integers: $f(x)=\frac{(x-11)(x-8)}{x^2-121}$.

Step:2

Express the denominator as a square: $f(x)=\frac{(x-11)(x-8)}{x^2-(11{)}^2}$.

Step:3

Factor the denominator as a difference of squares: $f(x)=\frac{(x-11)(x-8)}{(x+11)(x-11)}$.

Step:4

Simplify the fraction by canceling out common factors.

Step:4.1

Eliminate the common factor $(x-11)$: $f(x)=\frac{\cancel{(x-11)}(x-8)}{(x+11)\cancel{(x-11)}}$.

Step:4.2

Write the simplified function: $f(x)=\frac{x-8}{x+11}$.

Knowledge Notes:

The problem-solving process involves factoring a rational function. Here are the relevant knowledge points and detailed explanations:

1. Factoring Quadratic Expressions: The AC method is a technique used to factor quadratics of the form $a{x}^2+bx+c$. It involves finding two numbers that multiply to $ac$ and add to $b$. These numbers are then used to split the middle term and factor by grouping.

2. Difference of Squares: This is a pattern that allows us to factor expressions of the form $a^2-b^2$ into $(a+b)(a-b)$. In this problem, $x^2-121$ is recognized as a difference of squares since $121$ is a perfect square (${11}^2$).

3. Simplifying Rational Expressions: After factoring, if the numerator and denominator have common factors, they can be canceled out to simplify the expression. This is because $\frac{a}{b}=\frac{c}{d}$ if $ad=bc$ and $b,d\ne 0$.

4. Cancelling Common Factors: When a factor appears in both the numerator and the denominator of a fraction, it can be canceled out, as it does not affect the value of the fraction.

By applying these concepts, the original function is simplified to its lowest terms, resulting in a more manageable expression.