Question

1. error analysis describe and correct the error in simplifying the rational expression. \\(\frac{x^2 + 16x + 48}{x^2 + 8x + 16} = \frac{x^2 + 2x + 3}{x^2 + x + 1}\\) (with red cross and some markings on the original expression)

Explanation:

Step1: Identify the error

The error is that the simplification tried to cancel out terms (like 16x, 48, 8x, 16) directly from the numerator and denominator, but in rational expressions, we can only cancel out common factors (after factoring the numerator and denominator), not individual terms. So we need to factor both the numerator and the denominator first.

Step2: Factor the numerator

Factor \(x^{2}+16x + 48\). We need two numbers that multiply to 48 and add up to 16. The numbers are 12 and 4. So, \(x^{2}+16x + 48=(x + 12)(x+4)\)

Step3: Factor the denominator

Factor \(x^{2}+8x + 16\). This is a perfect square trinomial, since \(8x = 2\times x\times4\) and \(16 = 4^{2}\). So, \(x^{2}+8x + 16=(x + 4)^{2}\)

Step4: Simplify the rational expression

Now, substitute the factored forms into the original expression: \(\frac{(x + 12)(x + 4)}{(x + 4)^{2}}\). We can cancel out the common factor \((x + 4)\) (assuming \(x
eq - 4\)): \(\frac{(x + 12)\cancel{(x + 4)}}{(x + 4)\cancel{(x + 4)}}=\frac{x + 12}{x + 4}\)

Answer:

The error was canceling individual terms instead of factoring and canceling common factors. The correct simplification is \(\frac{x + 12}{x + 4}\) (for \(x
eq - 4\))