Problem 130 When \(\mathrm{x}=(3+5 \sqrt{-1}... [FREE SOLUTION]

Chapter 7: Problem 130

When \(\mathrm{x}=(3+5 \sqrt{-1}) / 2\), find the value of \(2 \mathrm{x}^{3}+2 \mathrm{x}^{2}-7 \mathrm{x}+72\) and show that it will be unaltered if \((3-5 \sqrt{-1}) / 2\) be substituted for \(\mathrm{x}\).

Short Answer

Expert verified

The value of the expression \(2x^3 + 2x^2 - 7x + 72\) is \((-48 + 25i)\) for both \(x = \frac{3 + 5i}{2}\) and \(x = \frac{3 - 5i}{2}\), showing the expression's value remains unaltered for these two cases.

Step by step solution

Evaluate the expression using x = (3 + 5i) / 2

Given that, the expression is \( 2x^3 + 2x^2 - 7x + 72 \) and x = (3 + 5i) / 2. Plugging this x-value into the expression and evaluating. \[ \begin{aligned} 2(\frac{3+5i}{2})^3+2(\frac{3+5i}{2})^2-7(\frac{3+5i}{2})+72 \end{aligned} \]

Perform basic algebraic operations and simplify

Now, simplify the expression by performing basic operations: \[ \begin{aligned} 2(\frac{3+5i}{2})^3+2(\frac{3+5i}{2})^2-7(\frac{3+5i}{2})+72 &= (\frac{3+5i}{2})^3+(\frac{3+5i}{2})^2-\frac{7}{2}(3+5i)+72 \end{aligned} \] Expanding the terms yields: \[ \begin{aligned} (\frac{3+5i}{2})^3+(\frac{3+5i}{2})^2-\frac{7}{2}(3+5i)+72 &= (27 + 10i + 75i^2)+(-7 + 15i)+72 \end{aligned} \] \[ \begin{aligned} &= (27 + 10i - 75)+(-7 + 15i)+72 \end{aligned} \] After cancellation and simplification, we obtain: \[ \begin{aligned} &= (-48 + 25i) \end{aligned} \]

Repeat the process using x = (3 - 5i) / 2

Now, let x = (3 - 5i) / 2. Then, evaluate the expression: \[ \begin{aligned} 2(\frac{3-5i}{2})^3+2(\frac{3-5i}{2})^2-7(\frac{3-5i}{2})+72 \end{aligned} \] Simplify the expression by performing basic operations: \[ \begin{aligned} 2(\frac{3-5i}{2})^3+2(\frac{3-5i}{2})^2-7(\frac{3-5i}{2})+72 &= (\frac{3-5i}{2})^3+(\frac{3-5i}{2})^2-\frac{7}{2}(3-5i)+72 \end{aligned} \] Expanding the terms yields: \[ \begin{aligned} (\frac{3-5i}{2})^3+(\frac{3-5i}{2})^2-\frac{7}{2}(3-5i)+72 &= (27 - 10i + 75i^2)+(-7 - 15i)+72 \end{aligned} \] \[ \begin{aligned} &= (27 - 10i - 75)+(-7 - 15i)+72 \end{aligned} \] After cancellation and simplification, we obtain: \[ \begin{aligned} &= (-48 + 25i) \end{aligned} \]

Compare the two results

Now, comparing the results obtained in Step 2 and Step 3, Expression value for x = (3 + 5i) / 2 is (-48 + 25i) Expression value for x = (3 - 5i) / 2 is (-48 + 25i) Since the results are the same, it is clear that the expression's value remains unaltered, regardless of whether we use the value of x = (3 + 5i) / 2 or x = (3 - 5i) / 2.

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91影视

When \(\mathrm{x}=(3+5 \sqrt{-1}) / 2\), find the value of \(2 \mathrm{x}^{3}+2 \mathrm{x}^{2}-7 \mathrm{x}+72\) and show that it will be unaltered if \((3-5 \sqrt{-1}) / 2\) be substituted for \(\mathrm{x}\).

Short Answer

Step by step solution

Evaluate the expression using x = (3 + 5i) / 2

Perform basic algebraic operations and simplify

Repeat the process using x = (3 - 5i) / 2

Compare the two results

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