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) = (x2 – 1){x2 – (1 + i + 1 – i)x + (1 – i)(1 + i)}
⇒ f(x) = (x2 – 1){x2 – 2x + (1 – i2)}
⇒ f(x) = (x2 – 1){x2 – 2x + (1 + 1)}
⇒ f(x) = (x2 – 1){x2 – 2x + 2}
⇒ f(x) = x2{x2 – 2x + 2} – {x2 – 2x + 2}
⇒ f(x) = x4 – 2x3 + 2x2 – x2 + 2x – 2
⇒ f(x) = x4 – 2x3 + x2 + 2x – 2
Thus, the standard form of the given polynomial is f(x) = x4 – 2x3 + x2 + 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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