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 Properties of the Polynomial
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Answer
Since 2+i is a zero and the coefficients of the polynomial are real, the complex conjugate 2−i must also be a zero.
Thus, the polynomial can be expressed as:
P(x)=(x−(2+i))(x−(2−i))(x−r)
where r is the third root.
Step 2
Expand the Polynomial
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Answer
We can first expand the factors related to the complex roots:
(x−(2+i))(x−(2−i))=(x−2)2+1=x2−4x+5
Now incorporating the third root:
P(x)=(x2−4x+5)(x−r)
Step 3
Match with Given Options
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Answer
Expanding this gives:
P(x)=x3−rx2−4x2+5x+4rx−5r
This matches with the polynomial form of four variants provided.
Now, for certain values of r, we can set k accordingly to match the coefficients.
The answer with the correct form following the above expansion is D: x3−5x2+kx+5.
