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. $x^3 - 4x^2 + kx + 5$
B. $x^3 - 4x^2 + kx + 5$
C. $x^3 - 5x^2 + kx + 5$
D. $x^3 - 5x^2... 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 polynomial roots
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Answer
Given that a polynomial has complex roots, if one root is 2+i, its conjugate 2−i must also be a root (as polynomial coefficients are real). Therefore, any polynomial that has both of these roots will include the factor (x−(2+i))(x−(2−i)).
Step 2
Calculate the polynomial from the roots
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Answer
The polynomial can be expressed as:
(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=x2−4x+4+1=x2−4x+5
Now, to find a polynomial that includes this quadratic factor, let's examine each option.
Step 3
Evaluate the options
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Answer
Among the options provided, we see that the polynomial option D is:
x3−5x2+kx+5
This polynomial can accommodate the quadratic factor we derived earlier when we expand it further with real coefficients. Thus, D is a suitable candidate.
Step 4
Conclusion on the polynomial
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Answer
Following this analysis, we conclude that the correct choice for a polynomial that has 2+i as a zero is:
