## Precalculus Textbook 8th Edition Section 2.5 Exercise 4 Solution

**Question**:

Write the polynomial f(x) = x(x – 1)(x – 1 – i)(x – 1 + i) in standard form, and identify the zeroes of the function and the x-intercept of its graph.

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**Solution**:

Given polynomial is f(x) = x(x – 1)(x – 1 – i)(x – 1 + i)

⇒ f(x) = x(x – 1){x – (1 + i)}{x – (1 – i)}

⇒ f(x) = (x^{2} – 1){x^{2} – (1 + i + 1 – i)x + (1 – i)(1 + i)}

⇒ f(x) = (x^{2} – 1){x^{2} – 2x + (1 – i^{2})}

⇒ f(x) = (x^{2} – 1){x^{2} – 2x + (1 + 1)}

⇒ f(x) = (x^{2} – 1){x^{2} – 2x + 2}

⇒ f(x) = x^{2}{x^{2} – 2x + 2} – {x^{2} – 2x + 2}

⇒ f(x) = x^{4} – 2x^{3} + 2x^{2} – x^{2} + 2x – 2

⇒ f(x) = x^{4} – 2x^{3} + x^{2} + 2x – 2

Thus, the standard form of the given polynomial is f(x) = x^{4} – 2x^{3} + x^{2} + 2x – 2.

Now, the zeroes of the given polynomial is

f(x) = x(x – 1)(x – 1 – i)(x – 1 + i) = 0

⇒ x = 0, 1, (1 + i) and (1 – i)

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