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Consider the polynomial $p(x) = ax^3 + bx^2 + cx - 6$ with $a$ and $b$ positive - HSC - SSCE Mathematics Extension 1 - Question 10 - 2016 - Paper 1
Question 10
Consider the polynomial $p(x) = ax^3 + bx^2 + cx - 6$ with $a$ and $b$ positive. Which graph could represent $p(x)$? (A) (B) (C) (D)
Worked Solution & Example Answer:Consider the polynomial $p(x) = ax^3 + bx^2 + cx - 6$ with $a$ and $b$ positive - HSC - SSCE Mathematics Extension 1 - Question 10 - 2016 - Paper 1
Step 1
Identify the characteristics of the polynomial
Answer
The given polynomial is a cubic polynomial. Since and are positive, the leading coefficient is positive. This implies that as approaches positive infinity, approaches positive infinity, and as approaches negative infinity, approaches negative infinity.
Step 2
Analyze the behavior of the graph
Answer
The graph of a cubic function typically has one of the following shapes: it can have one real root (crossing the x-axis once) or three real roots (crossing the x-axis three times). In this case, since both and are positive, it suggests that the polynomial could have a local minimum and maximum, further indicating that it does not cross the x-axis at the extremes and must yield a shape that looks 'W' or 'M'.
Step 3
Examine the options
Answer
Let's analyze the options:
- Graph (A): Shows a similar cubic shape that displays one intersection with the x-axis which relates to expected behavior at both ends.
- Graph (B): Looks like it has no minimum or maximum and doesn’t match the conditions.
- Graph (C): The graph appears to behave incorrectly regarding the polynomial characteristics.
- Graph (D): Similar to B, does not match the expected behavior.
The most fitting representation of in line with these properties is therefore graph (A).