Factor f(x)=(x^2-19x+88)/(x^2-121)
The question asks to perform a mathematical operation known as factoring on the given function f(x), which is expressed as a rational expression. This involves breaking down both the numerator (x^2-19x+88) and the denominator (x^2-121) into their simpler factors (products of polynomials that, when multiplied together, would give the original polynomial). Factoring can simplify the expression and possibly allow for the cancellation of common factors in both the numerator and denominator, thus simplifying the expression to its lowest terms. The problem also invites to see if there are any special factorizations like the difference of squares that apply, especially noticeable in the denominator’s structure.
$f \left(\right. x \left.\right) = \frac{x^{2} - 19 x + 88}{x^{2} - 121}$
Answer
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:
Factoring Quadratic Expressions: The AC method is a technique used to factor quadratics of the form $ax^2 + bx + c$. 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.
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 ($11^2$).
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 $b, d \neq 0$.
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.
