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
Determine the Conjugate Root
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Answer
Since 2+i is a root and the coefficients of the polynomial are real, its conjugate 2−i must also be a root.
Step 2
Form a Quadratic from the Roots
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Answer
We can form a quadratic polynomial from these roots:
(x−(2+i))(x−(2−i))=(x−2−i)(x−2+i)
Using the difference of squares, this expands to:
(x−2)2+1=x2−4x+4+1=x2−4x+5
Step 3
Multiply with Linear Factor
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Answer
To find a cubic polynomial, we can multiply the quadratic we found with a linear term (x−r), where r is real. Thus, the polynomial can be expressed as:
p(x)=k(x2−4x+5)
which expands to:
p(x)=kx3−4kx2+5k
Step 4
Check Each Polynomial Option
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Now, we can compare this form against the provided options:
A. x3−4x2+kx (doesn't match the form)
B. x3−4x2+kx+5 (has an extra constant term)
C. x3−5x2+kx (doesn't match the form)
D. x3−5x2+kx+5 (has an extra constant term)
None of these match exactly, but given that A represents a similar structure, we conclude:
