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
Given that f(x) is a monic polynomial of degree 3 with real coefficients, if 2 + i is one root, its conjugate 2 - i must also be a root. Therefore, the roots of f(x) are:
r1=3
r2=2+i
r3=2−i
Step 2
Construct the polynomial
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Answer
Using the roots, we can express the polynomial as:
f(x)=(x−r1)(x−r2)(x−r3)
Substituting the roots gives:
f(x)=(x−3)((x−2)−i)((x−2)+i)
The factors (x−2−i)(x−2+i) simplify to ((x−2)2+1), thus we have:
f(x)=(x−3)((x−2)2+1).
Expanding this yields:
f(x)=(x−3)(x2−4x+4+1)f(x)=(x−3)(x2−4x+5)
Using the distributive property, we can derive:
f(x)=x3−4x2+5x−3x2+12x−15f(x)=x3−7x2+17x−15
Step 3
Select the correct option
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Answer
From the expanded form, we can conclude:
f(x)=x3−7x2+17x−15
Thus, the correct option that matches is: B. f(x)=x3−7x2+17x−15.
