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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 properties of the polynomial

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Answer

The polynomial $p(x)$ is of degree 4, which means it can have at most 4 zeros. Since it has a repeated zero of multiplicity 2, we can represent this as $(x - a)^2$ for some constant $a$ .

Additionally, it is given that $p(x)$ is divisible by $x^2 + x + 1$ , which is a quadratic polynomial. The roots of this polynomial can be computed using the quadratic formula. Given that the discriminant is negative, it has two complex roots.

Thus, $p(x)$ can be expressed as:

$p(x) = (x - a)^2 (x^2 + x + 1).$

Step 2

Analyze the graph characteristics

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Answer

The graph of the polynomial will exhibit the following characteristics:

Since there is a repeated zero at $x = a$ , the graph must touch the x-axis at this point; the curve will not cross the x-axis here.
The polynomial is of even degree (4), which implies that the ends of the graph will tend to the same direction (either both upwards or both downwards) as $x o ightarrow ext{or} ightarrow - ext{infinity}$ .
The presence of two complex roots indicates that the polynomial does not have real x-intercepts corresponding to these roots but contributes to the overall 'wavy' nature of the function, affecting local maxima and minima.

Based on these characteristics, the correct option must show the curve touching the x-axis at one point and having both ends trending upwards.

Step 3

Select the correct graph

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Answer

From the options provided, option C is the only graph that matches the identified characteristics. It touches the x-axis at a single point (the repeated zero) without crossing it, and both ends of the graph extend upwards, consistent with a degree 4 polynomial.

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

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

Identify the properties of the polynomial

Analyze the graph characteristics

Select the correct graph

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