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A monic polynomial $p(x)$ of degree 4 has one repeated zero of multiplicity 2 and is divisible by $x^2 + x + 1$ - HSC - SSCE Mathematics Extension 1 - Question 5 - 2020 - Paper 1
Question 5
A monic polynomial $p(x)$ of degree 4 has one repeated zero of multiplicity 2 and is divisible by $x^2 + x + 1$. Which of the following could be the graph of $p(x)$... show full transcript
Worked Solution & Example Answer:A monic polynomial $p(x)$ of degree 4 has one repeated zero of multiplicity 2 and is divisible by $x^2 + x + 1$ - HSC - SSCE Mathematics Extension 1 - Question 5 - 2020 - Paper 1
Step 1
Identify the characteristics of the polynomial
Answer
The polynomial has the following characteristics:
- It is monic and has a degree of 4.
- It has a repeated zero of multiplicity 2, meaning it touches the x-axis at this zero and does not cross it.
- It is divisible by , which does not have real roots.
Hence, must contain these elements.
Step 2
Determine the implications for the graph
Answer
From the characteristics identified above, we can conclude:
- The graph must touch the x-axis at the repeated zero, indicating a local minimum or maximum at this point.
- There will be no x-intercepts from the terms related to , resulting in the graph remaining above or below the x-axis at these points.
- Given that the degree is 4, the ends of the graph will go off in the same direction (either both up or both down).
Step 3
Analyze the given options
Answer
We analyze each option:
- Option A: The graph crosses the x-axis, which contradicts the repeated zero property.
- Option B: Similar to A, it also crosses the x-axis.
- Option C: The graph touches the x-axis, indicating a repeated zero and does not cross it. Additionally, it maintains end behavior consistent with a degree 4 polynomial.
- Option D: The graph exhibits similar crossing behavior as A and B.
Therefore, Option C is the only graph that satisfactorily represents the polynomial's characteristics.