Which polynomial could have $2 + i$ as a zero, given that $k$ is a real number?
A - HSC - SSCE Mathematics Extension 2 - Question 6 - 2021 - Paper 1
Question 6
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 6 - 2021 - Paper 1
Step 1
Identify the characteristics of the complex zero
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Answer
Since the polynomial has 2+i as a zero and k is a real number, its conjugate 2−i must also be a zero. Therefore, the polynomial must have the form of a quadratic factor (x−(2+i))(x−(2−i)).
Step 2
Expand the quadratic factor
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Answer
Expanding the quadratic factor:
(x−(2+i))(x−(2−i))=(x−2−i)(x−2+i)
Using the difference of squares:
(x−2)2−i2=(x−2)2+1
Now, expanding further gives:
(x2−4x+4+1)=x2−4x+5
Step 3
Construct the polynomial
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Answer
The polynomial can be constructed as:
P(x)=(x2−4x+5)(x−r)
where r is another root. We further examine the given options using polynomial long division or factor fitting.
Step 4
Evaluate the options based on coefficients
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Answer
After evaluating each option, the polynomial needs to align with the expanded form from earlier while ensuring real coefficients when combined with the linear term.
Evaluating:
Option A does not fit as it does not yield the necessary terms of the quadratic.
Option B is viable since it accounts for the constant of 5 which is necessary.
Options C and D can be dismissed as they also do not align.
