Question

error analysis in exercises 25 and 26, describe and correct the error in performing the operation. 25. (x² - 3x + 4)-(x³ + 7x - 2) = x² - 3x + 4 - x³ + 7x - 2 = -x³ + x² + 4x + 2 26. (2x - 7)³=(2x)³ - 7³ = 8x³ - 343

Explanation:

Step1: Analyze Exercise 25 error

When distributing the negative sign in \((x^{2}-3x + 4)-(x^{3}+7x - 2)\), the sign of each term in \((x^{3}+7x - 2)\) should be changed. The error is that the sign of \(7x\) and \(-2\) was not changed correctly.
\((x^{2}-3x + 4)-(x^{3}+7x - 2)=x^{2}-3x + 4 - x^{3}-7x + 2\)

Step2: Simplify the corrected expression in Exercise 25

Combine like - terms: \(x^{2}-x^{3}+(-3x-7x)+(4 + 2)=-x^{3}+x^{2}-10x + 6\)

Step3: Analyze Exercise 26 error

The formula for \((a - b)^{3}=a^{3}-3a^{2}b + 3ab^{2}-b^{3}\), not \(a^{3}-b^{3}\). Here \(a = 2x\) and \(b = 7\).
\((2x-7)^{3}=(2x)^{3}-3\times(2x)^{2}\times7+3\times(2x)\times7^{2}-7^{3}\)

Step4: Simplify the corrected expression in Exercise 26

\((2x)^{3}-3\times(2x)^{2}\times7+3\times(2x)\times7^{2}-7^{3}=8x^{3}-3\times4x^{2}\times7 + 3\times2x\times49-343=8x^{3}-84x^{2}+294x - 343\)

Answer:

For Exercise 25: The error was incorrect sign - distribution. The correct result is \(-x^{3}+x^{2}-10x + 6\).
For Exercise 26: The error was using the wrong formula. The correct result is \(8x^{3}-84x^{2}+294x - 343\).