Simplify (x^4-8x^2+15)/(x^4-9)
The given problem asks to perform algebraic simplification on the provided rational expression. This involves reducing the expression (x^4 - 8x^2 + 15) / (x^4 - 9) to its simplest form by factoring both the numerator and the denominator, and then canceling out any common factors that appear in both to simplify the expression as much as possible. This process should yield a more concise expression which is algebraically equivalent to the original.
$\frac{x^{4} - 8 x^{2} + 15}{x^{4} - 9}$
Answer
Solution:
Step 1: Simplify the numerator.
Step 1.1
Express $x^4$ as $(x^2)^2$.
$$\frac{(x^2)^2 - 8x^2 + 15}{x^4 - 9}$$
Step 1.2
Introduce $u = x^2$ and replace $x^2$ with $u$.
$$\frac{u^2 - 8u + 15}{x^4 - 9}$$
Step 1.3
Factor the quadratic $u^2 - 8u + 15$ by searching for two numbers that multiply to $15$ and add up to $-8$.
Step 1.3.1
Identify the numbers as $-5$ and $-3$.
Step 1.3.2
Factor the quadratic to get $(u - 5)(u - 3)$.
$$\frac{(u - 5)(u - 3)}{x^4 - 9}$$
Step 1.4
Substitute $u$ back with $x^2$.
$$\frac{(x^2 - 5)(x^2 - 3)}{x^4 - 9}$$
Step 2: Simplify the denominator.
Step 2.1
Express $x^4$ as $(x^2)^2$.
$$\frac{(x^2 - 5)(x^2 - 3)}{(x^2)^2 - 9}$$
Step 2.2
Rewrite $9$ as $3^2$.
$$\frac{(x^2 - 5)(x^2 - 3)}{(x^2)^2 - 3^2}$$
Step 2.3
Factor using the difference of squares formula $a^2 - b^2 = (a + b)(a - b)$, where $a = x^2$ and $b = 3$.
$$\frac{(x^2 - 5)(x^2 - 3)}{(x^2 + 3)(x^2 - 3)}$$
Step 3: Cancel the common factor.
Step 3.1
Eliminate the common term $(x^2 - 3)$.
$$\frac{(x^2 - 5) \cancel{(x^2 - 3)}}{(x^2 + 3) \cancel{(x^2 - 3)}}$$
Step 3.2
Simplify the expression to get the final result.
$$\frac{x^2 - 5}{x^2 + 3}$$
Knowledge Notes:
Factoring Quadratics: The process of breaking down a quadratic equation into the product of two binomials. The AC method involves finding two numbers that multiply to give the product of the leading coefficient and the constant term (AC), and add up to the middle coefficient (B).
Substitution: A technique used to simplify expressions or equations by replacing a variable with another expression or variable.
Difference of Squares: A pattern used in algebra where $a^2 - b^2$ can be factored into $(a + b)(a - b)$.
Cancelling Common Factors: In a fraction, if the numerator and denominator share a common factor, it can be cancelled out to simplify the expression.
Expressions and Equations: An expression is a combination of symbols that represents a value, whereas an equation is a statement that asserts the equality of two expressions.
Latex Formatting: A typesetting system that is widely used for mathematical and scientific documents due to its capability to render complex mathematical expressions clearly.
