Simplify (z^2-5z+6)/(z^2-4)
In this problem, you are asked to perform algebraic simplification on a rational expression. The expression provided consists of a numerator, which is a quadratic polynomial (z² - 5z + 6), and a denominator, which is another quadratic polynomial (z² - 4). Simplifying the expression typically involves factoring the polynomials if possible and then cancelling out any common factors that appear in both the numerator and the denominator. The goal of the simplification process is to rewrite the expression in its simplest form, which often makes it easier to analyze or work with.
$\frac{z^{2} - 5 z + 6}{z^{2} - 4}$
Answer
Solution:
Step 1: Factor the numerator
Step 1.1: Apply the AC method
Identify two numbers that multiply to give $c$ and add up to $b$ in the quadratic expression $x^2 + bx + c$. Here, we need two numbers that multiply to $6$ and add to $-5$. These numbers are $-3$ and $-2$.
Step 1.2: Express the factored form
The factored form of the numerator is $\frac{(z - 3)(z - 2)}{z^2 - 4}$.
Step 2: Factor the denominator
Step 2.1: Express $4$ as a square
Rewrite $4$ as the square of $2$, resulting in $\frac{(z - 3)(z - 2)}{z^2 - 2^2}$.
Step 2.2: Factor using the difference of squares
Utilize the difference of squares formula $a^2 - b^2 = (a + b)(a - b)$, where $a = z$ and $b = 2$, to factor the denominator as $(z + 2)(z - 2)$.
Step 3: Simplify the expression
Step 3.1: Eliminate the common factor
Remove the common factor $(z - 2)$ from both the numerator and the denominator.
Step 3.2: Final simplified form
The simplified expression is $\frac{z - 3}{z + 2}$.
Knowledge Notes:
To solve the given problem, the following knowledge points are relevant:
Factoring Quadratics: The process of breaking down a quadratic expression into a product of two binomials. The AC method is one approach where you look for two numbers that multiply to give the product of the coefficient of $x^2$ (A) and the constant term (C), and that also add up to the coefficient of $x$ (B).
Difference of Squares: A special factoring formula that states $a^2 - b^2 = (a + b)(a - b)$. This is used when both terms in a binomial are perfect squares separated by a subtraction sign.
Simplifying Rational Expressions: The process of reducing a fraction to its simplest form by canceling out common factors from the numerator and the denominator.
Cancellation Law for Fractions: If a factor appears in both the numerator and the denominator of a fraction, it can be canceled out, as long as it's not zero.
Latex Formatting: Mathematical expressions are formatted in Latex to improve readability and to present the solution in a clear and professional manner. For example, the quadratic expression $z^2 - 5z + 6$ is written in Latex as $z^2 - 5z + 6$.
By applying these concepts, we can simplify the given rational expression by factoring both the numerator and the denominator and then canceling out the common terms.
