#### (4, 3)Question: Let $ f(x) = rac{x^4 - 10x^2 + 25}{x^2 - 5} $. Simplify $ f(x) $ and determine the value of $ f(2) $.
["Simplify the Function and Evaluate at $ x = 2 $: A Step-by-Step Guide", "When working with rational functions like $ f(x) = \frac{x^4 - 10x^2 + 25}{x^2 - 5} $, simplifying the expression first is key to making computation easier and uncovering deeper mathematical insights. In this article, we’ll simplify $ f(x) $ and then compute $ f(2) $, addressing the common question: Let $ f(x) = \frac{x^4 - 10x^2 + 25}{x^2 - 5} $. Simplify $ f(x) $ and determine the value of $ f(2) $.", "---", "### Step 1: Analyze the Numerator and Denominator", "The function is:", "$$\nf(x) = \frac{x^4 - 10x^2 + 25}{x^2 - 5}\n$$", "Observe the numerator: $ x^4 - 10x^2 + 25 $. This resembles a perfect square trinomial in terms of $ x^2 $. Recall:", "$$\n(a - b)^2 = a^2 - 2ab + b^2\n$$", "Let $ a = x^2 $, $ b = 5 $. Then:", "$$\nx^4 - 10x^2 + 25 = (x^2 - 5)^2\n$$", "So the function becomes:", "$$\nf(x) = \frac{(x^2 - 5)^2}{x^2 - 5}\n$$", "---", "### Step 2: Simplify the Expression", "As long as $ x^2 - 5 <br/>\neq 0 $ (i.e., $ x <br/>\neq \pm\sqrt{5} $), we can cancel one factor:", "$$\nf(x) = x^2 - 5 \quad \ ext{for } x <br/>\neq \pm\sqrt{5}\n$$", "This expression simplifies cleanly, but it’s important to note the domain restriction due to the original denominator.", "---", "### Step 3: Evaluate $ f(2) $", "Now compute $ f(2) $ using the simplified expression (and confirm validity):", "$$\nf(2) = (2)^2 - 5 = 4 - 5 = -1\n$$", "Since $ 2 <br/>\neq \pm\sqrt{5} $, the cancellation is valid.", "---", "### Why This Simplification Matters", "Simplifying rational functions like $ f(x) $ helps avoid unnecessary computation and reveals structural properties. Though the original expression looks complicated, recognizing $ x^4 - 10x^2 + 25 = (x^2 - 5)^2 $ transforms it into a basic quadratic form — ideal for evaluation and graphing.", "---", "### Final Answer", "After simplifying:", "$$\nf(x) = x^2 - 5, \quad x <br/>\neq \pm\sqrt{5}\n$$", "and evaluating at $ x = 2 $:", "$$\nf(2) = -1\n$$", "This streamlined approach ensures accuracy, efficiency, and deeper understanding — essential for mastering function simplification and evaluation.", "---", "Keywords: simplify rational function, $ f(x) = \frac{x^4 - 10x^2 + 25}{x^2 - 5} $, evaluate $ f(2) $, algebraic simplification, domain exclusion."]
