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
The roots of the polynomial are 3, 2 + i, and since the coefficients are real, the complex conjugate 2 - i must also be a root.
Step 2
Construct the polynomial
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Answer
Using the roots, we can write the polynomial as:
f(x)=(x−3)(x−(2+i))(x−(2−i))
First, let's simplify the complex roots:
(x−(2+i))(x−(2−i))=(x−2−i)(x−2+i)=(x−2)2+1=x2−4x+5
Now, we can expand:
f(x)=(x−3)(x2−4x+5)
Expanding this gives:
=x3−4x2+5x−3x2+12x−15=x3−7x2+17x−15
Step 3
Compare with options
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Answer
Upon comparing, we can see that:
Option A: x3−7x2−17x+15 (not a match)
Option B: x3−7x2+17x−15 (matches the derived polynomial)
