Which polynomial could have $2 + i$ as a zero, given that $k$ is a real number?
A - HSC - SSCE Mathematics Extension 2 - Question 8 - 2021 - Paper 1
Question 8
Which polynomial could have $2 + i$ as a zero, given that $k$ is a real number?
A. $x^3 - 4x^2 + kx$
B. $x^3 - 4x^2 + kx + 5$
C. $x^3 - 5x^2 + kx$
D. $x^3 - 5x^2 + k... show full transcript
Worked Solution & Example Answer:Which polynomial could have $2 + i$ as a zero, given that $k$ is a real number?
A - HSC - SSCE Mathematics Extension 2 - Question 8 - 2021 - Paper 1
Step 1
Identify the conjugate zero
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Answer
Since the coefficients of the polynomial must be real, if 2+i is a zero, then its conjugate 2−i must also be a zero.
Step 2
Form the polynomial using known zeros
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Answer
The polynomial can be expressed as:
f(x)=(x−(2+i))(x−(2−i))(x−r)
Where r is another real root. The product of the conjugate pair can be calculated as follows:
(x−(2+i))(x−(2−i))=(x−2)2+1=x2−4x+5
Step 3
Evaluate the polynomial forms
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Answer
Given that we need k to be arbitrary while still allowing the polynomial's roots to maintain their properties, option B is the correct polynomial. Therefore the polynomial matching the conditions would be option D, as it allows for both conjugate zeros.
