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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 $p(x)$?
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 is monic (leading coefficient is 1) and has a total degree of 4. It has a repeated zero of multiplicity 2, meaning that at this point, the graph must touch the x-axis but not cross it. The polynomial is also divisible by , which introduces complex roots, contributing to the composition of the final function.
Step 2
Analyze the potential graphs
Answer
Given the polynomial's degree and properties, we need to filter the options according to the characteristics we identified. The graph must show one double root (the repeated zero) where the curve merely touches the x-axis. We also should consider that there will be a pair of complex conjugate roots since does not intersect the x-axis.
Step 3
Select the correct graph
Answer
Upon examining the options, Graph C meets all the criteria mentioned: it displays a double root where the curve touches the x-axis, and the overall shape is consistent with the remaining degree of the polynomial, exhibiting the required upturns and downturns.