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 $p(x)$?
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 Characteristics of the Polynomial
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Answer
The polynomial p(x)=ax3+bx2+cx−6 is a cubic polynomial since its highest degree is 3. Given that a>0 and b>0, we can determine the overall behavior of the graph in the interval of positive and negative values for x.
Step 2
Behavior of the Graph as $x o ext{infinity}$ and $x o - ext{infinity}$
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Answer
As xoextinfinity, the term ax3 dominates, leading the graph to rise to extinfinity. As xo−extinfinity, ax3 also dominates, making the graph fall to −extinfinity. This suggests that the ends of the graph will point upwards.
Step 3
Determine the Roots of the Polynomial
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Answer
The polynomial has at least one real root. This can be inferred from the Intermediate Value Theorem, which states that since the polynomial is continuous, it must cross the x-axis at least once.
Step 4
Choose the Correct Graph
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Answer
Given that the polynomial has the described behaviors, the graph that correctly represents this behavior is Graph (A), which starts low for negative x-values, rises to cross the x-axis, and ends high for positive x-values.
