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Topic 1.4 · Polynomial Definition
What is the leading term, degree, and leading coefficient of a polynomial p(x)?
p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ··· + a₀
n is the highest exponent
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Topic 1.4 · Example 1b
What is the leading coefficient and degree of y = 12x − 7x³ + 11?
Find the term with the HIGHEST exponent first
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Topic 1.4 · Relative Extrema
What is a relative (local) minimum? What is a relative (local) maximum?
Think about direction changes
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Topic 1.4 · Absolute Extrema
How do you find the absolute (global) maximum and minimum from a graph?
Compare all local values
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Topic 1.4 · Fun Facts
What are the two 'Fun Facts' about polynomials regarding zeros and even-degree polynomials?
Between zeros · even degree
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Topic 1.4 · Points of Inflection
What is a point of inflection? What happens to the rate of change at a point of inflection?
Think about concavity changing
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Topic 1.4 · Example 3 — Intervals
On (−4,−3), g(x) is concave up and DECREASING. Is the ROC of g increasing or decreasing? Why?
Concave up = ROC is... even while going downhill
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Topic 1.4 · Example 4
C(t) = −1.37t⁵ + 4.218t⁴ − 0.357t² + 3 on [0,3]. At what time t does the number of cars change from increasing to decreasing?
Look for the local maximum
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✓ Example 1b — y = 12x − 7x³ + 11
LC = −7 · Degree = 3
The leading term is −7x³ (highest power). The negative sign belongs to the coefficient.
Don't be fooled by 12x appearing first — it has degree 1, not 3.
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✓ Polynomial Vocabulary
Leading term: aₙxⁿ · Degree: n · LC: aₙ
Leading term = term with highest power of x.
Degree = that highest exponent n.
Leading coefficient = the coefficient aₙ of that term.
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✓ Absolute Extrema
Abs Max = greatest of all local maxima · Abs Min = least of all local minima
Check all local max values → greatest = absolute max.
Check all local min values → least = absolute min.
If the function extends to ±∞, write N/A.
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✓ Relative Extrema
Rel Min: decreasing → increasing (valley) · Rel Max: increasing → decreasing (peak)
Relative (local) min: where function switches from decreasing to increasing.
Relative (local) max: where function switches from increasing to decreasing.
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✓ Points of Inflection
Where concavity changes (up→down or down→up) · ROC transitions (inc→dec or dec→inc)
At an inflection point, the concavity flips. This means the ROC changes direction — from increasing to decreasing or vice versa. NOT an extremum of the function itself.
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✓ Fun Facts About Polynomials
1) Between 2 zeros → at least one local max or min · 2) Even degree → global max OR global min
Fact 1: Between any 2 real zeros there must be a local extremum.
Fact 2: Even-degree polynomials (unrestricted domain) always have a global max or global min.
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✓ Example 4 — Parking Lot
t ≈ 2.4456 hours
The cars switch from increasing to decreasing at a local maximum. Use a graphing calculator to find the max of C(t) on [0,3] → t ≈ 2.4456.
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✓ ROC on (−4,−3) — Concave Up + Decreasing
ROC is INCREASING
Concave up = slope is getting larger. Going downhill but getting less steep → negative slopes become less negative → ROC increases.
Key: concavity and direction are independent!