A monic polynomial, $f(x)$, of degree 3 with real coefficients has 3 and $2 + i$ as two of its roots - HSC - SSCE Mathematics Extension 2 - Question 4 - 2024 - Paper 1
Question 4
A monic polynomial, $f(x)$, of degree 3 with real coefficients has 3 and $2 + i$ as two of its roots.
Which of the following could be $f(x)$?
A. $f(x) = x^3 - 7x^2... show full transcript
Worked Solution & Example Answer:A monic polynomial, $f(x)$, of degree 3 with real coefficients has 3 and $2 + i$ as two of its roots - HSC - SSCE Mathematics Extension 2 - Question 4 - 2024 - Paper 1
Step 1
Identify the Roots
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Answer
Since the polynomial has real coefficients and one of the roots is the complex number 2+i, its conjugate 2−i must also be a root. Therefore, the roots of the polynomial f(x) are 3, 2+i, and 2−i.
Step 2
Form the Polynomial
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Answer
To construct the polynomial f(x), we will use the fact that it can be expressed as:
f(x)=(x−3)((x−(2+i))(x−(2−i)))
The quadratic factor can be calculated as:
(x−(2+i))(x−(2−i))=(x−2)2+1=x2−4x+5.
Thus, the polynomial can be rewritten as:
f(x)=(x−3)(x2−4x+5).
Step 3
Expand the Polynomial
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Answer
Next, we can expand this expression:
f(x)=(x−3)(x2−4x+5)
