(a) Prove that \(\sqrt{23}\) is irrational - HSC - SSCE Mathematics Extension 2 - Question 12 - 2023 - Paper 1
Question 12
(a) Prove that \(\sqrt{23}\) is irrational.
(b) Prove that for all real numbers \(x\) and \(y\), where \(x^2 + y^2 \neq 0\),
\[\frac{(x+y)^2}{x^2 + y^2} \leq 2.\]
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Worked Solution & Example Answer:(a) Prove that \(\sqrt{23}\) is irrational - HSC - SSCE Mathematics Extension 2 - Question 12 - 2023 - Paper 1
Step 1
Find the remaining zeros of the polynomial \(P(z)\).
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Answer
Knowing that (2 + i) and (2 - i) are roots, we can express (P(z)
) as follows:
P(z)=(z−(2+i))(z−(2−i))(z−α)(z−β).
Calculating the product for the first two factors:
(z−(2+i))(z−(2−i))=(z−2)2+1=z2−4z+5.
Thus we can rewrite (P(z)) as:
P(z)=(z2−4z+5)(z2+az+b).
From here we will either complete the square or apply synthetic division to further factor to find the remaining roots.
