Simplify (y^2-3y-4)/(y^2-6y+8)
The question is asking to perform the algebraic operation of simplification on a rational expression. The expression provided is a fraction where both the numerator and the denominator are polynomials—a quadratic trinomial in both cases. The task is to simplify this complex fraction to its simplest form that typically may involve factoring both the numerator and the denominator to see if there are any common factors that can be cancelled out.
$\frac{y^{2} - 3 y - 4}{y^{2} - 6 y + 8}$
Answer
Solution:
Step 1 Apply the AC method to factorize the numerator $y^2 - 3y - 4$.
Step 1.1 Identify two integers whose product equals $c$ and sum equals $b$ for the quadratic $x^2 + bx + c$. Here, find integers with a product of $-4$ and a sum of $-3$. The integers are $-4$ and $1$.
Step 1.2 Express the numerator in its factored form using the found integers: $\frac{(y - 4)(y + 1)}{y^2 - 6y + 8}$.
Step 2 Factor the denominator $y^2 - 6y + 8$ using the AC method.
Step 2.1 For the quadratic $x^2 + bx + c$, find two integers whose product is $c$ and sum is $b$. Here, we need integers with a product of $8$ and a sum of $-6$. The integers are $-4$ and $-2$.
Step 2.2 Write the denominator in its factored form using these integers: $\frac{(y - 4)(y + 1)}{(y - 4)(y - 2)}$.
Step 3 Eliminate the common factor $(y - 4)$.
Step 3.1 Remove the common factor from both the numerator and the denominator: $\frac{\cancel{(y - 4)}(y + 1)}{\cancel{(y - 4)}(y - 2)}$.
Step 3.2 Simplify the expression to its reduced form: $\frac{y + 1}{y - 2}$.
Knowledge Notes:
Factoring Quadratics: The process of breaking down a quadratic equation into the product of two binomials. The AC method is a technique used to factor quadratics of the form $ax^2 + bx + c$ where $a$, $b$, and $c$ are integers. It involves finding two numbers that multiply to $ac$ and add up to $b$.
AC Method: This method is particularly useful when the leading coefficient (the coefficient of $x^2$) is not 1. However, in the given problem, the leading coefficient is 1, which simplifies the process.
Canceling Common Factors: When a common factor is present in both the numerator and 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 of simplifying rational expressions involves factoring both the numerator and the denominator and then canceling out common factors. The goal is to write the expression in its simplest form.
Quadratic Factoring Patterns: Recognizing patterns in quadratic expressions can speed up the factoring process. For example, if $c$ is negative in $x^2 + bx + c$, we look for one positive and one negative integer in the factoring process. If $c$ is positive and $b$ is negative, both integers used for factoring will be negative.
